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<p><strong>Description:</strong> Covers the application of graph theory to communication networks, surveying their configuration, topology, and optimization.</p> <p><strong>Speaker:</strong> Marten van Dijk</p>
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<p><span m="390">The</span> <span m="510">following</span> <span m="950">content</span> <span m="1550">is</span> <span m="1670">provided</span> <span m="2110">under</span> <span m="2390">a</span> <span m="2430">Creative</span> <span m="2830">Commons</span> <span m="3230">license.</span> <span m="4340">Your</span> <span m="4530">support</span> <span m="5030">will</span> <span m="5200">help</span> <span m="5440">MIT</span> <span m="5890">OpenCourseWare</span> <span m="6680">continue</span> <span m="7190">to</span> <span m="7270">offer</span> <span m="7690">high-quality</span> <span m="8440">educational</span> <span m="9070">resources</span> <span m="9700">for</span> <span m="9850">free.</span> <span m="11050">To</span> <span m="11060">make</span> <span m="11260">a</span> <span m="11310">donation</span> <span m="11990">or</span> <span m="12260">view</span> <span m="12700">additional</span> <span m="13120">materials</span> <span m="13670">from</span> <span m="13810">hundreds</span> <span m="14250">of</span> <span m="14360">MIT</span> <span m="14780">courses,</span> <span m="15900">visit</span> <span m="16110">MIT</span> <span m="16530">OpenCourseWare</span> <span m="17580">at</span> <span m="17750">ocw.mit.edu.</span> </p>
<p><span m="23010">MARTEN VAN DIJK: So</span> <span m="23280">today,</span> <span m="23600">we're</span> <span m="23770">going</span> <span m="23970">to</span> <span m="24090">talk</span> <span m="24390">about</span> <span m="25310">communication</span> <span m="26000">networks.</span> <span m="28030">Communication</span> <span m="28610">networks</span> <span m="29100">is</span> <span m="29240">a</span> <span m="29280">great</span> <span m="29600">application</span> <span m="30270">of</span> <span m="30320">graph</span> <span m="31160">theory.</span> <span m="32020">So</span> <span m="32570">what</span> <span m="32710">we're</span> <span m="32830">going</span> <span m="33020">to</span> <span m="33120">study</span> <span m="33560">is,</span> <span m="34720">how</span> <span m="34910">do</span> <span m="35000">you</span> <span m="35180">route</span> <span m="35930">packets</span> <span m="36600">through</span> <span m="36890">networks?</span> <span m="38030">So</span> <span m="38210">you</span> <span m="38310">have</span> <span m="38720">the</span> <span m="38890">internet,</span> <span m="39330">which</span> <span m="39660">is a</span> <span m="39990">chaotic</span> <span m="40430">network.</span> <span m="40900">It's</span> <span m="41180">not</span> <span m="41520">organized.</span> <span m="42850">We</span> <span m="42990">are</span> <span m="43050">interested</span> <span m="43730">in</span> <span m="43860">highly</span> <span m="44190">structured</span> <span m="44990">networks</span> <span m="46270">and</span> <span m="46880">you</span> <span m="47030">can</span> <span m="47160">find</span> <span m="47460">them,</span> <span m="47610">for</span> <span m="47770">example,</span> <span m="48400">in</span> <span m="48710">parallel</span> <span m="49380">computers,</span> <span m="49910">where</span> <span m="50100">you</span> <span m="50260">want</span> <span m="50440">to</span> <span m="50580">route</span> <span m="52820">the</span> <span m="52980">data</span> <span m="53230">flow.</span> <span m="53980">You</span> <span m="54170">can</span> <span m="54390">find</span> <span m="54680">them</span> <span m="54880">in</span> <span m="56140">certain</span> <span m="56810">telephone</span> <span m="57370">switches</span> <span m="60230">networks</span> <span m="61170">and</span> <span m="61360">so</span> <span m="61550">on.</span> </p>
<p><span m="62620">So</span> <span m="63400">we are</span> <span m="63560">going</span> <span m="63730">to</span> <span m="63840">talk</span> <span m="64080">about</span> <span m="65290">a</span> <span m="65470">few</span> <span m="65600">very</span> <span m="65840">special</span> <span m="66330">ones,</span> <span m="66920">binary</span> <span m="67390">trees,</span> <span m="67780">and</span> <span m="67960">then</span> <span m="68100">slowly</span> <span m="69080">we will</span> <span m="69290">figure</span> <span m="69600">out</span> <span m="69870">what</span> <span m="70160">all</span> <span m="70410">these</span> <span m="71050">performance</span> <span m="71560">measures</span> <span m="71950">really</span> <span m="72200">mean.</span> <span m="72960">This</span> <span m="73120">one</span> <span m="73290">has</span> <span m="73390">to</span> <span m="73480">do</span> <span m="73580">with</span> <span m="73700">latency.</span> <span m="74400">We</span> <span m="74520">have</span> <span m="74690">switches,</span> <span m="75340">their</span> <span m="75480">size,</span> <span m="76010">the</span> <span m="76080">number</span> <span m="76390">of</span> <span m="76530">them,</span> <span m="77300">congestion,</span> <span m="78690">and</span> <span m="78940">then</span> <span m="79090">we will</span> <span m="79270">slowly</span> <span m="80230">get</span> <span m="80570">down</span> <span m="80860">to</span> <span m="81140">Benes</span> <span m="81480">network,</span> <span m="82090">which</span> <span m="82260">is</span> <span m="82430">a</span> <span m="82800">really</span> <span m="83040">beautiful</span> <span m="83470">network</span> <span m="83940">with</span> <span m="84170">beautiful</span> <span m="85700">parameters.</span> <span m="86770">And</span> <span m="86970">we</span> <span m="87050">are going</span> <span m="87270">to</span> <span m="87340">prove</span> <span m="87650">those.</span> </p>
<p><span m="89510">So</span> <span m="92110">let's</span> <span m="92290">start</span> <span m="92530">off</span> <span m="92780">with</span> <span m="93100">the</span> <span m="94010">first</span> <span m="94380">one,</span> <span m="94700">the</span> <span m="94810">complete</span> <span m="95230">binary</span> <span m="95620">tree,</span> <span m="96530">and</span> <span m="96770">let</span> <span m="96920">me</span> <span m="96990">draw</span> <span m="97260">it</span> <span m="97400">for</span> <span m="97620">you.</span> <span m="104400">In</span> <span m="104590">this</span> <span m="105360">network,</span> <span m="106130">we</span> <span m="106350">will</span> <span m="106570">have</span> <span m="108650">a</span> <span m="108810">root</span> <span m="109930">and</span> <span m="110120">let me</span> <span m="110620">just</span> <span m="110820">draw</span> <span m="111150">it</span> <span m="111420">first</span> <span m="112060">We</span> <span m="114220">have</span> <span m="115280">vertices</span> <span m="116440">that</span> <span m="117410">represent</span> <span m="117840">here</span> <span m="118270">the</span> <span m="118350">switches.</span> <span m="119300">So</span> <span m="119490">these</span> <span m="119680">circles--</span> <span m="120330">let</span> <span m="120530">me</span> <span m="121460">explain</span> <span m="121980">it</span> <span m="122070">over</span> <span m="122360">here--</span> <span m="123930">actually</span> <span m="124370">represent</span> <span m="125740">a</span> <span m="125850">switch.</span> </p>
<p><span m="127120">And</span> <span m="127720">the</span> <span m="127870">idea</span> <span m="128259">is</span> <span m="128560">that</span> <span m="129120">these</span> <span m="129650">actually</span> <span m="130070">direct</span> <span m="133490">packets</span> <span m="135040">through</span> <span m="135770">the</span> <span m="135850">network.</span> <span m="141660">And</span> <span m="141970">these</span> <span m="142190">packets</span> <span m="142600">are</span> <span m="142890">fixed-size</span> <span m="145120">packets</span> <span m="145430">of</span> <span m="145550">data,</span> <span m="145920">so</span> <span m="146150">like,</span> <span m="146420">I don't</span> <span m="146560">know,</span> <span m="146910">say</span> <span m="147590">4,000</span> <span m="149090">bytes</span> <span m="149990">or</span> <span m="150200">bits</span> <span m="150870">or</span> <span m="151030">whatever</span> <span m="151580">the</span> <span m="151690">network</span> <span m="152340">wants</span> <span m="152600">you</span> <span m="152740">to</span> <span m="153230">comply</span> <span m="153640">to.</span> <span m="154140">So</span> <span m="154290">these</span> <span m="154520">are</span> <span m="154590">fixed-size</span> <span m="155940">pieces</span> <span m="156370">of</span> <span m="156490">data.</span> <span m="158090">So</span> <span m="158240">what</span> <span m="158360">we</span> <span m="158640">want</span> <span m="158750">is we</span> <span m="158870">want</span> <span m="159140">to</span> <span m="159750">be</span> <span m="159870">able</span> <span m="160490">from</span> <span m="161040">every</span> <span m="161330">terminal--</span> <span m="162050">and</span> <span m="162180">the</span> <span m="162250">terminal</span> <span m="162790">I</span> <span m="162930">will</span> <span m="163180">denote</span> <span m="163710">by</span> <span m="164380">a</span> <span m="164480">square--</span> <span m="165500">from</span> <span m="165760">every</span> <span m="166020">terminal,</span> <span m="166980">I</span> <span m="167160">want</span> <span m="167400">to</span> <span m="167530">be</span> <span m="167710">able</span> <span m="168940">to</span> <span m="170700">reach</span> <span m="171110">any</span> <span m="171380">other</span> <span m="171610">terminal.</span> </p>
<p><span m="173030">So</span> <span m="173250">what</span> <span m="173400">is</span> <span m="173500">a</span> <span m="173580">terminal?</span> <span m="174040">A terminal</span> <span m="174470">is</span> <span m="174600">like</span> <span m="174920">a</span> <span m="175180">computer</span> <span m="176260">or</span> <span m="176380">something</span> <span m="176710">like</span> <span m="176920">that.</span> <span m="178610">It's</span> <span m="179150">actually</span> <span m="179490">the</span> <span m="179590">source</span> <span m="180050">and the</span> <span m="180280">destination</span> <span m="183890">of</span> <span m="184610">data.</span> <span m="192880">So</span> <span m="194640">what</span> <span m="194830">we are</span> <span m="194970">looking</span> <span m="195280">for</span> <span m="195660">is</span> <span m="195910">how</span> <span m="196070">can</span> <span m="196250">we</span> <span m="196410">route--</span> <span m="196840">how we</span> <span m="197050">can</span> <span m="197190">find</span> <span m="197440">a</span> <span m="197580">network</span> <span m="198280">of</span> <span m="198430">switches</span> <span m="199960">that</span> <span m="200190">are</span> <span m="200270">connected</span> <span m="200670">through</span> <span m="200900">wires,</span> <span m="201530">fibers,</span> <span m="202230">or--</span> <span m="202570">yeah?</span> <span m="202980">What's</span> <span m="203120">the</span> <span m="203170">question?</span> </p>
<p><span m="204129">AUDIENCE: Can you</span> <span m="204608">move down</span> <span m="205087">a bit</span> <span m="205566">the</span> <span m="206045">top of</span> <span m="206524">the--</span> <span m="207003">how it's</span> <span m="207482">getting cut off?</span> <span m="207961">No,</span> <span m="208440">the--</span> </p>
<p><span m="208919">That one.</span> </p>
<p><span m="209877">MARTEN VAN DIJK: Oh,</span> <span m="210356">sorry.</span> </p>
<p><span m="210835">AUDIENCE: All right.</span> <span m="212272">Thank you.</span> </p>
<p><span m="214200">MARTEN VAN DIJK: So</span> <span m="214430">what</span> <span m="214710">we want</span> <span m="214990">is</span> <span m="215140">we</span> <span m="215310">want</span> <span m="215480">to</span> <span m="215950">route</span> <span m="217630">packets</span> <span m="218080">that's</span> <span m="218360">come</span> <span m="218550">from</span> <span m="218800">any</span> <span m="219030">terminal</span> <span m="219670">to</span> <span m="219850">any</span> <span m="220070">other</span> <span m="220240">terminal.</span> <span m="221320">That is</span> <span m="221650">what</span> <span m="221880">our</span> <span m="222080">goal</span> <span m="222410">is</span> <span m="222650">and we</span> <span m="222700">want</span> <span m="222930">to</span> <span m="224200">make</span> <span m="224460">sure</span> <span m="224730">that</span> <span m="224870">that</span> <span m="225160">is</span> <span m="225770">efficient.</span> <span m="227080">So</span> <span m="227320">the</span> <span m="227420">first</span> <span m="227920">one</span> <span m="228150">is</span> <span m="228340">this</span> <span m="228510">binary</span> <span m="228890">tree.</span> <span m="229790">And</span> <span m="230550">let's</span> <span m="233610">see</span> <span m="233970">how</span> <span m="234160">this</span> <span m="234380">may</span> <span m="234600">work.</span> </p>
<p><span m="235170">We</span> <span m="235320">may</span> <span m="235550">have</span> <span m="235850">switches</span> <span m="237380">that</span> <span m="238020">actually</span> <span m="244890">have</span> <span m="246060">inputs</span> <span m="246650">coming</span> <span m="247040">from</span> <span m="247840">terminals.</span> <span m="256910">And</span> <span m="258320">the</span> <span m="258519">switches</span> <span m="258820">may</span> <span m="258990">also</span> <span m="259300">output</span> <span m="259690">to</span> <span m="259820">terminals,</span> <span m="261060">so</span> <span m="261120">here</span> <span m="261320">at</span> <span m="261399">the</span> <span m="261470">bottom.</span> <span m="262260">At</span> <span m="262410">this</span> <span m="262610">site,</span> <span m="262840">we</span> <span m="262940">have</span> <span m="263110">a</span> <span m="263160">similar</span> <span m="264990">structure.</span> <span m="265780">this</span> <span m="265960">is</span> <span m="266100">the</span> <span m="266210">root</span> <span m="266400">of</span> <span m="266620">the tree.</span> <span m="267510">We</span> <span m="267660">have</span> <span m="267790">another</span> <span m="268010">switch</span> <span m="268390">over</span> <span m="268660">here.</span> <span m="269500">We</span> <span m="269670">go</span> <span m="269830">down,</span> <span m="270380">we</span> <span m="270540">go</span> <span m="270690">up</span> <span m="271000">here,</span> <span m="272510">and</span> <span m="272710">once</span> <span m="272980">more,</span> <span m="273610">like</span> <span m="273840">this.</span> <span m="275260">And</span> <span m="275360">again,</span> <span m="275780">we</span> <span m="275830">have--</span> <span m="277040">oops.</span> <span m="278950">We</span> <span m="279130">have</span> <span m="279760">input</span> <span m="280110">coming</span> <span m="280420">in</span> <span m="280690">or</span> <span m="281070">an</span> <span m="281180">output</span> <span m="281520">coming</span> <span m="281820">out</span> <span m="283430">to</span> <span m="283560">their</span> <span m="283730">respective</span> <span m="284330">terminals.</span> </p>
<p><span m="289120">So</span> <span m="290410">what</span> <span m="290560">is</span> <span m="290680">happening</span> <span m="291060">here</span> <span m="291400">is</span> <span m="291730">that</span> <span m="292800">I</span> <span m="292910">would</span> <span m="293140">like</span> <span m="293480">to</span> <span m="294040">have</span> <span m="294690">an</span> <span m="295420">input--</span> <span m="297260">say</span> <span m="297360">input</span> <span m="297690">zero</span> <span m="299120">wants</span> <span m="299570">to</span> <span m="300000">travel</span> <span m="300450">all</span> <span m="300680">the</span> <span m="300790">way</span> <span m="301000">over</span> <span m="301380">to</span> <span m="301520">say,</span> <span m="301830">the</span> <span m="302020">output</span> <span m="302470">that</span> <span m="302780">is</span> <span m="303480">present</span> <span m="303830">over</span> <span m="304100">here.</span> <span m="304280">So</span> <span m="304410">let</span> <span m="304510">me</span> <span m="304640">label</span> <span m="304950">these.</span> <span m="305800">So</span> <span m="306010">we</span> <span m="306120">have</span> <span m="306640">the</span> <span m="306990">output</span> <span m="308170">zero,</span> <span m="309580">input</span> <span m="309960">one,</span> <span m="311670">output</span> <span m="312030">one,</span> <span m="314170">input</span> <span m="315150">two</span> <span m="315860">and</span> <span m="316320">output</span> <span m="317590">two,</span> <span m="319450">input</span> <span m="320360">three,</span> <span m="321150">and</span> <span m="321420">output</span> <span m="322080">four.</span> <span m="323940">So</span> <span m="326480">well,</span> <span m="326640">I</span> <span m="326700">can definitely</span> <span m="327220">reach</span> <span m="328050">every</span> <span m="328380">single</span> <span m="328700">output</span> <span m="329100">from</span> <span m="329310">any</span> <span m="329570">input</span> <span m="330240">so</span> <span m="330400">that's</span> <span m="330660">great.</span> </p>
<p><span m="331470">So</span> <span m="331610">this</span> <span m="331840">looks</span> <span m="332050">like</span> <span m="333120">something</span> <span m="333590">that</span> <span m="333790">you</span> <span m="333980">are</span> <span m="334080">familiar</span> <span m="334560">with,</span> <span m="334800">right?</span> <span m="335040">It's</span> <span m="335540">just</span> <span m="335890">a</span> <span m="336050">tree.</span> <span m="340910">It's</span> <span m="341040">a</span> <span m="341080">directed</span> <span m="341460">graph,</span> <span m="342000">but</span> <span m="342170">these</span> <span m="342340">edges</span> <span m="342870">go</span> <span m="343400">in</span> <span m="343560">both</span> <span m="343840">directions,</span> <span m="344560">right?</span> <span m="344870">So</span> <span m="345060">I</span> <span m="345170">have an</span> <span m="345490">edge that</span> <span m="345790">goes</span> <span m="346000">from</span> <span m="346120">here</span> <span m="346310">to</span> <span m="346450">here</span> <span m="346710">and</span> <span m="346940">back</span> <span m="348070">from</span> <span m="348350">here</span> <span m="348620">to</span> <span m="348790">here.</span> <span m="349750">So</span> <span m="350960">this</span> <span m="351140">is</span> <span m="351280">the</span> <span m="351360">kind</span> <span m="351620">of</span> <span m="352050">layout</span> <span m="352460">that</span> <span m="352560">you</span> <span m="353020">could</span> <span m="353260">try</span> <span m="353610">out</span> <span m="353840">first</span> <span m="354500">to</span> <span m="354610">see</span> <span m="354860">whether</span> <span m="355120">this</span> <span m="355340">type</span> <span m="355560">of</span> <span m="355700">network</span> <span m="356130">would</span> <span m="356260">lead to</span> <span m="356500">good</span> <span m="356660">performance.</span> <span m="358440">So</span> <span m="358600">let's</span> <span m="359140">have</span> <span m="359340">a</span> <span m="359450">look</span> <span m="359930">at</span> <span m="360090">the</span> <span m="360180">different</span> <span m="360540">parameters</span> <span m="361720">and</span> <span m="361950">see</span> <span m="363370">how</span> <span m="363580">well</span> <span m="363820">this</span> <span m="364040">behaves.</span> <span m="364670">So</span> <span m="365520">here,</span> <span m="365700">we</span> <span m="365790">have</span> <span m="366010">a</span> <span m="366060">few</span> <span m="366240">parameters</span> <span m="367160">that</span> <span m="367930">we will</span> <span m="368190">be</span> <span m="368320">talking</span> <span m="368690">about.</span> </p>
<p><span m="371860">So</span> <span m="372080">first</span> <span m="372410">of</span> <span m="372550">all,</span> <span m="372810">let's</span> <span m="372960">talk</span> <span m="373220">about</span> <span m="374560">the</span> <span m="374690">latency</span> <span m="376390">in</span> <span m="376590">this</span> <span m="376770">particular</span> <span m="377180">network.</span> <span m="378690">So</span> <span m="378800">how</span> <span m="378990">are</span> <span m="379090">we</span> <span m="379110">going</span> <span m="379420">to</span> <span m="379600">measure</span> <span m="379940">this?</span> <span m="381680">Well,</span> <span m="382410">we're</span> <span m="382530">going</span> <span m="382770">to</span> <span m="382910">look</span> <span m="383100">at</span> <span m="383150">this</span> <span m="383330">graph</span> <span m="383870">and</span> <span m="384040">we're</span> <span m="384160">going</span> <span m="384470">to</span> <span m="384610">measure</span> <span m="384930">it</span> <span m="385070">by</span> <span m="385210">the</span> <span m="385300">number</span> <span m="385650">of</span> <span m="386190">wires</span> <span m="386650">that</span> <span m="386820">you</span> <span m="386900">need</span> <span m="387070">to</span> <span m="387160">go</span> <span m="387430">through</span> <span m="388090">from</span> <span m="388330">an</span> <span m="388400">input</span> <span m="388740">to</span> <span m="388860">an</span> <span m="388990">output.</span> <span m="390930">So</span> <span m="391020">let</span> <span m="391170">me</span> <span m="391260">write</span> <span m="391450">this</span> <span m="391620">down.</span> <span m="395590">So</span> <span m="395710">the</span> <span m="395800">latency</span> <span m="397390">is</span> <span m="397650">the</span> <span m="397740">time</span> <span m="398680">that is</span> <span m="398950">required</span> <span m="406780">for</span> <span m="407030">a</span> <span m="407090">packet</span> <span m="408520">to</span> <span m="408920">travel</span> <span m="415640">from</span> <span m="416000">an</span> <span m="416130">input</span> <span m="417520">to an</span> <span m="417770">output.</span> <span m="425830">And</span> <span m="426690">how</span> <span m="426940">are</span> <span m="427040">we</span> <span m="427060">going</span> <span m="427260">to</span> <span m="427370">measure</span> <span m="427650">this?</span> <span m="428170">Well,</span> <span m="428320">we're</span> <span m="428440">just going</span> <span m="428660">to</span> <span m="428750">measure</span> <span m="429120">this</span> <span m="429670">by</span> <span m="429810">the</span> <span m="429910">number</span> <span m="430610">of</span> <span m="430800">wires</span> <span m="431210">that</span> <span m="431300">we need</span> <span m="431560">to</span> <span m="431650">go</span> <span m="431870">through.</span> </p>
<p><span m="433010">So</span> <span m="433920">this</span> <span m="434180">you</span> <span m="434730">have</span> <span m="434860">seen</span> <span m="435080">before.</span> <span m="435510">We</span> <span m="435660">can</span> <span m="435790">measure</span> <span m="436030">this</span> <span m="436210">by</span> <span m="436330">the</span> <span m="436440">diameter</span> <span m="437280">of</span> <span m="437410">that</span> <span m="437530">particular</span> <span m="437940">graph.</span> <span m="439330">So</span> <span m="439770">here,</span> <span m="440060">we</span> <span m="440270">will</span> <span m="440450">define</span> <span m="440790">it</span> <span m="440900">for</span> <span m="441050">a</span> <span m="441090">network.</span> <span m="441720">So</span> <span m="441860">the</span> <span m="441980">diameter</span> <span m="442990">of a</span> <span m="443310">network</span> <span m="447930">is</span> <span m="448200">going</span> <span m="448420">to</span> <span m="448540">be</span> <span m="448720">the</span> <span m="448830">length</span> <span m="453300">of</span> <span m="453550">the</span> <span m="453630">shortest</span> <span m="454280">path</span> <span m="461210">between</span> <span m="465170">the</span> <span m="465270">input</span> <span m="466700">and</span> <span m="466940">output</span> <span m="474580">that</span> <span m="474750">are</span> <span m="474840">furthest</span> <span m="475360">apart.</span> <span m="476950">So</span> <span m="477170">let's</span> <span m="477340">have</span> <span m="477500">a</span> <span m="477540">look</span> <span m="477740">at</span> <span m="477860">the</span> <span m="479120">graph</span> <span m="479460">above.</span> </p>
<p><span m="482050">So</span> <span m="482330">for</span> <span m="482490">example,</span> <span m="484300">we</span> <span m="484500">can</span> <span m="484670">clearly</span> <span m="485010">see</span> <span m="485360">that,</span> <span m="485570">for</span> <span m="485660">example,</span> <span m="486530">input</span> <span m="487270">and</span> <span m="487480">output--</span> <span m="488680">so</span> <span m="488950">say,</span> <span m="489160">input</span> <span m="489460">zero</span> <span m="490220">and</span> <span m="490460">output</span> <span m="491380">one</span> <span m="492170">are</span> <span m="492330">connected</span> <span m="492980">by</span> <span m="493170">just</span> <span m="493480">going</span> <span m="493810">up</span> <span m="495000">one</span> <span m="495200">step</span> <span m="495560">over</span> <span m="495840">here,</span> <span m="500600">but</span> <span m="500720">just</span> <span m="500990">going</span> <span m="501320">up</span> <span m="502520">from</span> <span m="502780">here</span> <span m="503040">to</span> <span m="503250">here.</span> <span m="504370">Then,</span> <span m="504540">this</span> <span m="504730">switch</span> <span m="505130">forwards</span> <span m="506280">the</span> <span m="506410">packet</span> <span m="507100">to</span> <span m="507300">this</span> <span m="507500">switch.</span> <span m="507880">This</span> <span m="508000">switch</span> <span m="508950">reroutes</span> <span m="509100">it,</span> <span m="509180">forwards</span> <span m="510030">it</span> <span m="510520">over</span> <span m="510670">here,</span> <span m="511740">and</span> <span m="512010">then it</span> <span m="512090">goes</span> <span m="512309">back</span> <span m="512490">to</span> <span m="512610">the</span> <span m="512760">output,</span> <span m="513840">output</span> <span m="514130">one.</span> <span m="514760">So</span> <span m="514900">for</span> <span m="515030">example,</span> <span m="515590">this</span> <span m="515700">particular</span> <span m="516770">path</span> <span m="517145">only has</span> <span m="517520">1,</span> <span m="517799">2,</span> <span m="518130">3,</span> <span m="518549">4</span> <span m="519600">edges.</span> </p>
<p><span m="520770">And</span> <span m="522130">what we are</span> <span m="522270">interested</span> <span m="522630">in</span> <span m="523600">is</span> <span m="524490">sort</span> <span m="524680">of</span> <span m="524800">the</span> <span m="524920">worst-case</span> <span m="526580">time</span> <span m="526900">that</span> <span m="527050">it</span> <span m="527180">requires</span> <span m="527910">to</span> <span m="528050">go</span> <span m="528310">from</span> <span m="528600">an</span> <span m="528710">input</span> <span m="529000">to</span> <span m="529110">an</span> <span m="529230">output.</span> <span m="529700">So</span> <span m="529710">that</span> <span m="529860">means</span> <span m="530150">that</span> <span m="530270">we are</span> <span m="530420">interested</span> <span m="531050">in a</span> <span m="531210">diameter.</span> <span m="532020">And a</span> <span m="532110">diameter</span> <span m="532560">is</span> <span m="533880">in</span> <span m="534020">this</span> <span m="534270">case,</span> <span m="534630">well,</span> <span m="534760">the</span> <span m="534880">shortest</span> <span m="535500">path</span> <span m="536590">that</span> <span m="536750">you</span> <span m="536820">can</span> <span m="537040">find</span> <span m="538240">from</span> <span m="538540">an</span> <span m="538630">input</span> <span m="539090">to</span> <span m="540400">an</span> <span m="540610">output</span> <span m="541030">that</span> <span m="541140">are</span> <span m="541300">furthest</span> <span m="541810">apart.</span> <span m="542060">So</span> <span m="542220">what</span> <span m="542390">are</span> <span m="542510">those</span> <span m="542940">who</span> <span m="543090">are</span> <span m="543170">furthest</span> <span m="543590">apart?</span> </p>
<p><span m="544340">Well,</span> <span m="544840">of</span> <span m="544970">course,</span> <span m="545170">you</span> <span m="545250">would</span> <span m="545360">like</span> <span m="545610">to</span> <span m="545710">go</span> <span m="545960">through</span> <span m="546220">here,</span> <span m="546500">right?</span> <span m="546780">So if</span> <span m="547370">I</span> <span m="547510">connect</span> <span m="548050">the</span> <span m="548160">input</span> <span m="548500">zero</span> <span m="548870">to</span> <span m="548990">say,</span> <span m="549620">output</span> <span m="549990">four,</span> <span m="550695">I</span> <span m="550950">will</span> <span m="551100">need to</span> <span m="551300">go</span> <span m="552490">all</span> <span m="552730">the</span> <span m="552810">way</span> <span m="553020">up</span> <span m="553360">through</span> <span m="553580">the</span> <span m="553740">route</span> <span m="554450">down</span> <span m="554780">to</span> <span m="555260">the</span> <span m="555400">output.</span> <span m="556400">And</span> <span m="556510">how</span> <span m="556660">many</span> <span m="557490">edges</span> <span m="557790">do</span> <span m="557950">we</span> <span m="558050">see</span> <span m="558260">here?</span> <span m="558520">1,</span> <span m="558800">2,</span> <span m="559080">3,</span> <span m="559610">4,</span> <span m="559910">5,</span> <span m="560260">6--</span> <span m="561690">so</span> <span m="561860">in this</span> <span m="562040">example,</span> <span m="563530">we</span> <span m="563690">have</span> <span m="563860">a</span> <span m="563960">diameter</span> <span m="566980">that</span> <span m="567080">is</span> <span m="567210">equal</span> <span m="567450">to</span> <span m="567590">six.</span> </p>
<p><span m="568820">And</span> <span m="569190">in</span> <span m="569330">general,</span> <span m="570190">if</span> <span m="570410">you</span> <span m="570570">are</span> <span m="570720">looking</span> <span m="571230">at</span> <span m="574020">n</span> <span m="574240">times</span> <span m="574570">n</span> <span m="574760">networks,</span> <span m="575480">what</span> <span m="575590">does it</span> <span m="575690">mean?</span> <span m="577030">n</span> <span m="577400">is</span> <span m="577580">the</span> <span m="577660">number</span> <span m="578380">of</span> <span m="578530">inputs</span> <span m="579640">and</span> <span m="580050">n is</span> <span m="580200">also</span> <span m="580490">the</span> <span m="580590">number</span> <span m="580850">of</span> <span m="581000">outputs.</span> <span m="581390">So</span> <span m="581530">in</span> <span m="581600">this</span> <span m="581800">case,</span> <span m="582030">we</span> <span m="582120">have</span> <span m="582390">a</span> <span m="583370">four</span> <span m="584360">times--</span> <span m="585325">well,</span> <span m="585600">this</span> <span m="585760">is</span> <span m="585880">actually</span> <span m="586230">three</span> <span m="586700">over</span> <span m="586820">here--</span> <span m="588760">we</span> <span m="588940">have</span> <span m="590190">four</span> <span m="590450">inputs</span> <span m="590850">and</span> <span m="590950">four</span> <span m="591220">outputs.</span> <span m="592180">So</span> <span m="592380">this</span> <span m="592610">particular</span> <span m="593030">example</span> <span m="593430">depicted</span> <span m="593870">on</span> <span m="593980">the</span> <span m="594060">board</span> <span m="594780">is</span> <span m="595080">a</span> <span m="595456">four</span> <span m="595832">times</span> <span m="596210">four</span> <span m="596460">network.</span> </p>
<p><span m="597680">So</span> <span m="597810">if</span> <span m="597940">you</span> <span m="598010">generalize</span> <span m="598740">this</span> <span m="598950">for</span> <span m="599320">any</span> <span m="599570">size</span> <span m="600680">binary</span> <span m="601320">tree,</span> <span m="603730">say,</span> <span m="604150">an n</span> <span m="604350">times</span> <span m="604670">n</span> <span m="604840">network,</span> <span m="606060">then</span> <span m="608990">what's</span> <span m="609190">the</span> <span m="609260">diameter</span> <span m="609920">of</span> <span m="610690">such</span> <span m="611420">a</span> <span m="611460">general</span> <span m="611830">network?</span> <span m="612380">Well,</span> <span m="613540">if</span> <span m="613770">we</span> <span m="613870">have</span> <span m="614290">n</span> <span m="614790">inputs</span> <span m="615310">and n</span> <span m="615630">outputs,</span> <span m="616980">well,</span> <span m="617140">we</span> <span m="617240">have</span> <span m="617470">to</span> <span m="617630">go</span> <span m="617850">all</span> <span m="618050">the</span> <span m="618140">way</span> <span m="618370">up</span> <span m="619170">through</span> <span m="620430">towards</span> <span m="620800">the</span> <span m="620930">root</span> <span m="621160">and</span> <span m="621250">all</span> <span m="621400">the</span> <span m="621500">way</span> <span m="621680">down.</span> <span m="622270">So</span> <span m="622490">we</span> <span m="622590">actually</span> <span m="623480">count</span> <span m="624120">the</span> <span m="624230">length</span> <span m="624790">of a</span> <span m="625180">leaf</span> <span m="625860">to</span> <span m="626650">the</span> <span m="626800">root</span> <span m="627030">here</span> <span m="627420">twice.</span> <span m="628940">So</span> <span m="629150">in</span> <span m="629250">general,</span> <span m="629780">we</span> <span m="629900">have</span> <span m="630080">a</span> <span m="630180">diameter</span> <span m="630760">that</span> <span m="630910">looks</span> <span m="631130">like</span> <span m="631340">this.</span> <span m="633010">It's</span> <span m="634680">2</span> <span m="634930">times</span> <span m="636970">1</span> <span m="637290">plus</span> <span m="638050">the</span> <span m="638190">logarithm</span> <span m="638860">of</span> <span m="639060">n.</span> </p>
<p><span m="639740">So</span> <span m="639990">in</span> <span m="640060">this</span> <span m="641450">lecture,</span> <span m="642380">we</span> <span m="642410">will</span> <span m="642750">have</span> <span m="643170">n</span> <span m="644800">is</span> <span m="645020">going</span> <span m="645200">to</span> <span m="645290">be</span> <span m="645420">a</span> <span m="645520">power</span> <span m="645860">of</span> <span m="646090">2,</span> <span m="648550">just</span> <span m="648800">to</span> <span m="648870">make</span> <span m="649060">calculations</span> <span m="649960">simple.</span> <span m="651030">And</span> <span m="652440">the</span> <span m="652560">logarithm</span> <span m="653050">is</span> <span m="653180">always</span> <span m="653450">to</span> <span m="653570">the</span> <span m="653660">base</span> <span m="653930">two.</span> <span m="656170">So</span> <span m="656350">this</span> <span m="656700">is</span> <span m="656850">a</span> <span m="656990">diameter</span> <span m="657300">of</span> <span m="657420">a</span> <span m="657540">general</span> <span m="659260">binary</span> <span m="659730">tree.</span> </p>
<p><span m="661330">And</span> <span m="663730">well,</span> <span m="663880">what</span> <span m="664060">are</span> <span m="664150">the</span> <span m="664260">other</span> <span m="664780">parameters?</span> <span m="665460">So</span> <span m="665620">that</span> <span m="665780">does</span> <span m="665990">not</span> <span m="666150">look</span> <span m="666330">too</span> <span m="666550">bad.</span> <span m="667320">It's</span> <span m="667630">logarithmic</span> <span m="668200">in</span> <span m="668410">answer.</span> <span m="668700">That</span> <span m="668870">sounds</span> <span m="669130">pretty</span> <span m="669380">good.</span> <span m="670420">What</span> <span m="670690">about</span> <span m="671510">the</span> <span m="671620">switch</span> <span m="672290">sizes?</span> <span m="674010">Well,</span> <span m="675230">how</span> <span m="675410">do</span> <span m="675590">I</span> <span m="675650">measure</span> <span m="676020">those?</span> <span m="676400">It's</span> <span m="676590">like</span> <span m="676770">the</span> <span m="676850">number</span> <span m="677130">of</span> <span m="677290">inputs</span> <span m="677740">that</span> <span m="677850">get</span> <span m="678120">into</span> <span m="678490">it</span> <span m="678770">and</span> <span m="678880">the</span> <span m="678920">number</span> <span m="679110">of</span> <span m="679190">outputs</span> <span m="679630">that</span> <span m="679780">get</span> <span m="679970">out.</span> <span m="680240">So</span> <span m="681150">in</span> <span m="681260">this</span> <span m="681510">case,</span> <span m="681920">I</span> <span m="682040">will</span> <span m="682210">have</span> <span m="683020">1,</span> <span m="683710">2</span> <span m="683960">inputs</span> <span m="684430">that</span> <span m="684600">go</span> <span m="684870">into</span> <span m="684990">this</span> <span m="685200">switch</span> <span m="686280">and</span> <span m="686430">there</span> <span m="686530">are</span> <span m="686690">two</span> <span m="687110">outputs</span> <span m="687620">coming</span> <span m="687940">out.</span> <span m="688150">So</span> <span m="688260">this</span> <span m="688470">is</span> <span m="688680">what</span> <span m="688840">we</span> <span m="688940">call</span> <span m="689160">a</span> <span m="689300">two</span> <span m="689490">times</span> <span m="689910">two</span> <span m="690140">switch.</span> <span m="691290">So</span> <span m="691520">this</span> <span m="691730">will</span> <span m="691850">be</span> <span m="692040">a two</span> <span m="692240">times</span> <span m="692590">two</span> <span m="692780">switch.</span> </p>
<p><span m="693600">But</span> <span m="693640">if</span> <span m="693750">you</span> <span m="693880">look</span> <span m="694040">at</span> <span m="694110">this</span> <span m="694290">one,</span> <span m="694430">for</span> <span m="694570">example,</span> <span m="695630">we</span> <span m="695750">see</span> <span m="696540">one,</span> <span m="697120">two,</span> <span m="697830">three</span> <span m="698180">outgoing</span> <span m="698690">edges</span> <span m="699600">and</span> <span m="699900">three</span> <span m="700450">ingoing</span> <span m="701020">edges.</span> <span m="701950">So</span> <span m="702050">this</span> <span m="702240">is</span> <span m="702370">actually</span> <span m="702780">a three</span> <span m="703040">times</span> <span m="703350">three</span> <span m="703570">switch.</span> <span m="705480">And</span> <span m="707610">in a</span> <span m="707840">general</span> <span m="708740">binary</span> <span m="709140">tree,</span> <span m="709710">we</span> <span m="709870">will</span> <span m="709970">see</span> <span m="710280">that</span> <span m="710950">all</span> <span m="711310">these</span> <span m="711480">intermediate</span> <span m="712140">nodes</span> <span m="712280">over</span> <span m="712680">here,</span> <span m="712910">they are</span> <span m="713210">all</span> <span m="713480">three</span> <span m="713730">times</span> <span m="713990">three</span> <span m="714190">switches.</span> <span m="714660">So</span> <span m="718070">approximately</span> <span m="718750">half</span> <span m="719100">of</span> <span m="719240">the</span> <span m="719340">switches</span> <span m="720150">are</span> <span m="720370">actually</span> <span m="720990">three</span> <span m="721280">times</span> <span m="721870">three</span> <span m="722220">switches.</span> <span m="724300">So</span> <span m="724440">that's</span> <span m="724670">the</span> <span m="724720">switch</span> <span m="725180">size.</span> </p>
<p><span m="730350">Now,</span> <span m="730510">you</span> <span m="730650">may</span> <span m="730800">say,</span> <span m="731160">well,</span> <span m="733170">why</span> <span m="733370">don't</span> <span m="733610">I</span> <span m="733680">use</span> <span m="734010">a</span> <span m="734310">larger-sized</span> <span m="735630">switch?</span> <span m="736310">That</span> <span m="736570">would</span> <span m="736750">help</span> <span m="736970">me</span> <span m="737160">a</span> <span m="737230">lot,</span> <span m="737470">right?</span> <span m="738200">If</span> <span m="738390">I</span> <span m="738440">could</span> <span m="738800">use,</span> <span m="740490">say,</span> <span m="740780">a</span> <span m="740840">four</span> <span m="741110">times</span> <span m="741380">four</span> <span m="741640">switch,</span> <span m="743640">then</span> <span m="744400">I</span> <span m="744520">would</span> <span m="744730">be</span> <span m="744930">able</span> <span m="745790">to</span> <span m="745920">have</span> <span m="746130">more</span> <span m="746360">inputs</span> <span m="746780">coming</span> <span m="747080">in,</span> <span m="747200">more</span> <span m="747480">outputs</span> <span m="747910">coming</span> <span m="748200">out,</span> <span m="748870">and</span> <span m="749150">I</span> <span m="749190">can</span> <span m="749430">actually</span> <span m="750770">maybe</span> <span m="751150">use</span> <span m="751500">a</span> <span m="751690">ternary</span> <span m="752050">tree</span> <span m="752680">rather</span> <span m="752920">than</span> <span m="753190">a</span> <span m="753270">binary</span> <span m="753740">tree.</span> <span m="754070">In</span> <span m="754210">a</span> <span m="754260">binary</span> <span m="754610">tree,</span> <span m="754930">every</span> <span m="756360">note</span> <span m="756720">at</span> <span m="756810">the</span> <span m="756920">level</span> <span m="757260">has</span> <span m="757710">two</span> <span m="757940">children,</span> <span m="758440">right?</span> <span m="759230">But</span> <span m="759350">we</span> <span m="759490">could</span> <span m="759860">design</span> <span m="760350">a</span> <span m="760500">tree</span> <span m="760795">that</span> <span m="761090">has</span> <span m="761430">at</span> <span m="761600">every</span> <span m="761910">level</span> <span m="762240">three</span> <span m="762510">children.</span> <span m="764070">So</span> <span m="764210">then, they</span> <span m="764320">can</span> <span m="764590">use</span> <span m="764980">four</span> <span m="765160">times</span> <span m="765410">four</span> <span m="765640">switches.</span> </p>
<p><span m="766580">But if</span> <span m="766810">you</span> <span m="766920">do</span> <span m="767080">that,</span> <span m="768380">then</span> <span m="768590">the</span> <span m="768700">path</span> <span m="769230">from</span> <span m="770200">the</span> <span m="770680">leaf</span> <span m="771220">up</span> <span m="771400">to</span> <span m="771520">the</span> <span m="771680">root</span> <span m="771940">is</span> <span m="772050">getting</span> <span m="772280">shorter</span> <span m="773150">and</span> <span m="773510">the</span> <span m="773580">diameter</span> <span m="773950">gets</span> <span m="774160">smaller.</span> <span m="774780">So</span> <span m="774850">if</span> <span m="775020">I</span> <span m="775120">increase</span> <span m="776740">the</span> <span m="776860">switch</span> <span m="777350">size--</span> <span m="778530">so</span> <span m="778800">rather</span> <span m="779000">than</span> <span m="779200">three</span> <span m="779390">times</span> <span m="779690">three,</span> <span m="779900">we</span> <span m="780020">look</span> <span m="780190">at</span> <span m="780290">four</span> <span m="780510">times</span> <span m="780810">four</span> <span m="781145">or five</span> <span m="781480">times</span> <span m="781800">five,</span> <span m="782110">six</span> <span m="782420">times</span> <span m="782690">six</span> <span m="783350">and</span> <span m="783530">so</span> <span m="783710">on--</span> <span m="783820">then the</span> <span m="784040">diameter</span> <span m="785240">will</span> <span m="785300">actually</span> <span m="786180">reduce.</span> <span m="787410">So</span> <span m="788220">what</span> <span m="788390">about</span> <span m="788810">having</span> <span m="789130">a</span> <span m="789180">monster</span> <span m="789600">switch,</span> <span m="792150">like</span> <span m="792460">I</span> <span m="792510">have</span> <span m="792660">just</span> <span m="792950">one</span> <span m="793170">switch</span> <span m="793730">and</span> <span m="793830">I</span> <span m="793930">have</span> <span m="794220">my</span> <span m="794400">input</span> <span m="795330">zero</span> <span m="795810">all</span> <span m="796010">the</span> <span m="796100">way</span> <span m="796340">up</span> <span m="796500">to</span> <span m="796740">input</span> <span m="798370">n</span> <span m="798540">minus</span> <span m="798950">1</span> <span m="800330">and</span> <span m="800530">then</span> <span m="800720">I</span> <span m="800820">have</span> <span m="801350">my</span> <span m="801600">outputs</span> <span m="802120">on</span> <span m="802210">the</span> <span m="802320">other</span> <span m="802490">side?</span> </p>
<p><span m="808960">Well,</span> <span m="810120">of</span> <span m="810310">course,</span> <span m="810630">the</span> <span m="810820">switch</span> <span m="811320">size</span> <span m="811970">is</span> <span m="812370">n</span> <span m="812620">times</span> <span m="813060">n</span> <span m="814180">but</span> <span m="814310">the</span> <span m="814400">diameter</span> <span m="816400">is</span> <span m="816550">nothing,</span> <span m="816950">right?</span> <span m="820400">The</span> <span m="820550">diameter</span> <span m="821040">is</span> <span m="821560">reduced</span> <span m="821950">to</span> <span m="822030">one.</span> <span m="822170">You</span> <span m="822260">can</span> <span m="822360">immediately</span> <span m="822720">go</span> <span m="822950">from</span> <span m="823180">an</span> <span m="823260">input</span> <span m="823530">to</span> <span m="823640">an</span> <span m="823740">output</span> <span m="824230">through</span> <span m="824440">the</span> <span m="824540">switch.</span> <span m="825610">But</span> <span m="825680">this,</span> <span m="825860">of</span> <span m="825970">course,</span> <span m="826200">conceals</span> <span m="826650">the</span> <span m="826730">problem.</span> <span m="827240">So</span> <span m="827440">what</span> <span m="827580">we are</span> <span m="827750">interested</span> <span m="828460">in</span> <span m="828610">is,</span> <span m="829380">well,</span> <span m="830420">we're</span> <span m="830510">actually</span> <span m="830780">really</span> <span m="831030">interested</span> <span m="832550">in</span> <span m="833680">how</span> <span m="833960">to</span> <span m="834130">solve</span> <span m="834620">the</span> <span m="834720">problem</span> <span m="835200">of</span> <span m="835470">routing</span> <span m="835990">all</span> <span m="836180">these</span> <span m="836380">inputs</span> <span m="836900">to</span> <span m="837050">these</span> <span m="837290">outputs</span> <span m="838290">using</span> <span m="841340">smaller</span> <span m="842280">switches</span> <span m="843150">of</span> <span m="843250">size</span> <span m="843660">three</span> <span m="843860">times</span> <span m="844140">three</span> <span m="844390">or</span> <span m="844490">two</span> <span m="844660">times</span> <span m="845020">two.</span> </p>
<p><span m="845800">What</span> <span m="845940">we're</span> <span m="846070">really</span> <span m="846280">interested</span> <span m="846810">in</span> <span m="847040">is,</span> <span m="847990">what</span> <span m="848180">is</span> <span m="848460">the</span> <span m="849220">internal</span> <span m="849770">structure</span> <span m="850350">in</span> <span m="850650">this</span> <span m="851070">monster</span> <span m="851470">switch?</span> <span m="851800">I</span> <span m="851850">sort</span> <span m="852250">of</span> <span m="852480">have</span> <span m="852840">concealed</span> <span m="853230">the</span> <span m="853320">problem</span> <span m="853790">by</span> <span m="853950">just</span> <span m="854070">saying,</span> <span m="854300">oh,</span> <span m="854450">I've</span> <span m="854660">got</span> <span m="854750">a</span> <span m="854820">big</span> <span m="855020">switch.</span> <span m="855990">But</span> <span m="858260">what</span> <span m="858430">we</span> <span m="858560">want</span> <span m="858730">to</span> <span m="858820">solve</span> <span m="859130">today</span> <span m="859520">is</span> <span m="860190">how</span> <span m="860440">do</span> <span m="860570">we</span> <span m="860710">do</span> <span m="860850">the</span> <span m="860980">routing</span> <span m="862240">in</span> <span m="862590">this</span> <span m="862860">case</span> <span m="863740">within</span> <span m="864290">the</span> <span m="864370">monster</span> <span m="864700">switch?</span> <span m="865640">So</span> <span m="866520">we</span> <span m="866650">want</span> <span m="866810">to</span> <span m="866910">use</span> <span m="868170">just</span> <span m="868700">small</span> <span m="869830">switch</span> <span m="870170">sizes</span> <span m="870780">and</span> <span m="870910">build</span> <span m="871730">up</span> <span m="872020">a</span> <span m="872080">network</span> <span m="872660">using</span> <span m="872970">these</span> <span m="873130">smaller</span> <span m="873570">ones,</span> <span m="873920">like</span> <span m="874140">three</span> <span m="874330">times</span> <span m="874590">three</span> <span m="875520">switches</span> <span m="875960">or</span> <span m="876090">two</span> <span m="876230">times</span> <span m="876560">two</span> <span m="876750">switches.</span> </p>
<p><span m="879950">Now,</span> <span m="885220">so</span> <span m="886390">that</span> <span m="887590">brings</span> <span m="887880">us</span> <span m="888100">to</span> <span m="888390">yet</span> <span m="888580">another</span> <span m="888880">parameter,</span> <span m="889990">because</span> <span m="890310">here,</span> <span m="890400">we'd</span> <span m="890560">like</span> <span m="890740">to</span> <span m="890900">count</span> <span m="891240">the</span> <span m="891330">number</span> <span m="891700">or</span> <span m="891800">smaller</span> <span m="892240">switches</span> <span m="892700">that</span> <span m="892880">we</span> <span m="892990">use</span> <span m="893850">and</span> <span m="894010">that</span> <span m="894180">relates</span> <span m="894490">to</span> <span m="894570">the</span> <span m="894700">cost</span> <span m="895120">of</span> <span m="895300">the</span> <span m="895390">network,</span> <span m="897620">the</span> <span m="897750">amount</span> <span m="898030">of</span> <span m="898100">hardware</span> <span m="898570">that</span> <span m="898670">you</span> <span m="898760">need</span> <span m="898910">to</span> <span m="899010">put</span> <span m="899180">into</span> <span m="899490">it.</span> <span m="900100">So</span> <span m="900280">in</span> <span m="900340">this</span> <span m="900520">example,</span> <span m="901700">we</span> <span m="901890">have</span> <span m="902880">the</span> <span m="902980">switch</span> <span m="905800">count.</span> <span m="909570">Well, it's</span> <span m="909880">pretty</span> <span m="910110">simple,</span> <span m="910560">right?</span> <span m="910810">It's</span> <span m="911420">1,</span> <span m="911914">2,</span> <span m="912408">3, 4,</span> <span m="912902">5, 6,</span> <span m="913396">7--</span> <span m="914880">we</span> <span m="915010">have</span> <span m="915200">seven</span> <span m="915610">switches.</span> </p>
<p><span m="916920">And</span> <span m="917460">in</span> <span m="917630">general,</span> <span m="919980">if we</span> <span m="920330">have</span> <span m="921350">n</span> <span m="921610">inputs--</span> <span m="923140">so</span> <span m="923640">1,</span> <span m="923960">2,</span> <span m="924190">3,</span> <span m="924410">4</span> <span m="924760">inputs--</span> <span m="925840">then</span> <span m="926020">the</span> <span m="926100">number</span> <span m="926410">of</span> <span m="926580">switches</span> <span m="927030">that we</span> <span m="927300">use</span> <span m="927610">in</span> <span m="927700">the</span> <span m="927750">binary</span> <span m="928140">tree</span> <span m="928600">is</span> <span m="928880">2</span> <span m="929170">times</span> <span m="929900">the</span> <span m="930000">number</span> <span m="930220">of</span> <span m="930340">inputs</span> <span m="931170">minus</span> <span m="931490">1.</span> <span m="933680">So</span> <span m="933930">let's</span> <span m="934130">write</span> <span m="934320">that</span> <span m="934440">down.</span> <span m="936590">So</span> <span m="936860">over</span> <span m="937010">here,</span> <span m="937250">we</span> <span m="937360">would</span> <span m="937520">have</span> <span m="938370">2</span> <span m="938590">times</span> <span m="939030">n</span> <span m="939500">minus</span> <span m="939930">1,</span> <span m="940220">which</span> <span m="940400">is</span> <span m="940500">the</span> <span m="940570">number</span> <span m="940770">of</span> <span m="940870">switches</span> <span m="941290">that you</span> <span m="941590">actually</span> <span m="941900">use.</span> </p>
<p><span m="945140">So</span> <span m="945390">how</span> <span m="945570">can</span> <span m="945710">you</span> <span m="945830">see</span> <span m="946080">that</span> <span m="946270">actually?</span> <span m="947360">So</span> <span m="947540">in</span> <span m="947620">general,</span> <span m="948080">we</span> <span m="948250">have</span> <span m="949760">1</span> <span m="950090">plus</span> <span m="950530">2</span> <span m="950930">plus</span> <span m="951300">4</span> <span m="952190">plus</span> <span m="952670">8</span> <span m="953640">and</span> <span m="953840">so</span> <span m="954100">on</span> <span m="954410">plus</span> <span m="956220">n.</span> <span m="957310">And</span> <span m="957510">it's</span> <span m="957650">a</span> <span m="957700">power</span> <span m="958070">of</span> <span m="958250">2,</span> <span m="958860">according to</span> <span m="959190">our</span> <span m="959430">assumptions.</span> <span m="960650">And</span> <span m="960800">if</span> <span m="960910">you</span> <span m="961030">add</span> <span m="961270">them</span> <span m="961450">all</span> <span m="961610">up,</span> <span m="962462">I</span> <span m="962890">think</span> <span m="963100">you'll--</span> <span m="964510">well,</span> <span m="964690">you</span> <span m="964770">can</span> <span m="964910">check</span> <span m="965110">for</span> <span m="965250">yourself</span> <span m="965710">that</span> <span m="965760">this</span> <span m="965930">is</span> <span m="966110">actually</span> <span m="966760">equal</span> <span m="967010">to</span> <span m="967160">2</span> <span m="967340">times</span> <span m="967690">n</span> <span m="967850">minus</span> <span m="968190">1.</span> </p>
<p><span m="971440">So</span> <span m="971610">now,</span> <span m="971820">we</span> <span m="972010">have</span> <span m="972600">the</span> <span m="972980">switches.</span> <span m="973260">So</span> <span m="973330">so</span> <span m="973960">far,</span> <span m="974250">this</span> <span m="974420">looks</span> <span m="974770">pretty</span> <span m="975050">good,</span> <span m="975320">actually.</span> <span m="976280">We</span> <span m="976470">use</span> <span m="976680">small</span> <span m="977350">switch</span> <span m="977670">sizes.</span> <span m="978060">The number</span> <span m="978570">of</span> <span m="978630">switches</span> <span m="979110">is</span> <span m="979470">linear</span> <span m="979910">in</span> <span m="980090">n.</span> <span m="981440">The</span> <span m="981610">diameter</span> <span m="982070">is logarithmic</span> <span m="982570">in</span> <span m="982890">n</span> <span m="983150">so</span> <span m="983300">that</span> <span m="983430">sounds</span> <span m="983740">good.</span> </p>
<p><span m="984430">So</span> <span m="984580">what</span> <span m="984710">about</span> <span m="985010">congestion?</span> <span m="985780">Do you</span> <span m="986120">any</span> <span m="986380">idea--</span> <span m="987800">what's</span> <span m="988730">the</span> <span m="988820">problem</span> <span m="989210">with</span> <span m="989360">this</span> <span m="989530">graph?</span> <span m="990850">What is</span> <span m="991020">the</span> <span m="991680">big,</span> <span m="992200">big</span> <span m="992650">problem</span> <span m="993090">here?</span> <span m="994700">What</span> <span m="994850">can</span> <span m="995070">happen</span> <span m="995910">in</span> <span m="996040">a</span> <span m="996260">very</span> <span m="997580">sort of</span> <span m="997900">worst-case</span> <span m="998410">scenario</span> <span m="999040">where</span> <span m="999590">the</span> <span m="999710">packets</span> <span m="1000200">get</span> <span m="1000350">routed</span> <span m="1000950">from</span> <span m="1001210">inputs</span> <span m="1001560">to the</span> <span m="1001710">outputs?</span> <span m="1002150">If</span> <span m="1003110">they</span> <span m="1003210">need</span> <span m="1003450">to</span> <span m="1003550">go</span> <span m="1003740">to</span> <span m="1003870">certain</span> <span m="1004160">locations,</span> <span m="1006400">then</span> <span m="1006540">they</span> <span m="1006610">all</span> <span m="1007620">may</span> <span m="1007860">have</span> <span m="1008140">to</span> <span m="1008250">travel</span> <span m="1008670">through</span> <span m="1008830">the</span> <span m="1009000">root.</span> <span m="1010160">So</span> <span m="1010770">you</span> <span m="1010910">get</span> <span m="1011080">congestion</span> <span m="1011760">over</span> <span m="1012030">here.</span> <span m="1012620">We</span> <span m="1012740">don't</span> <span m="1012950">like</span> <span m="1013210">that.</span> <span m="1014190">So</span> <span m="1014380">this</span> <span m="1014740">root</span> <span m="1015040">is</span> <span m="1015190">actually</span> <span m="1015480">then</span> <span m="1015670">overloaded.</span> </p>
<p><span m="1016270">Actually,</span> <span m="1016630">you can</span> <span m="1016780">already</span> <span m="1017050">see</span> <span m="1017290">that</span> <span m="1018060">say,</span> <span m="1019640">this</span> <span m="1019900">particular</span> <span m="1020300">switch--</span> <span m="1020780">if</span> <span m="1020890">this</span> <span m="1020980">switch</span> <span m="1021740">fails,</span> <span m="1022960">then</span> <span m="1023930">actually,</span> <span m="1024500">we</span> <span m="1024640">will</span> <span m="1024790">have</span> <span m="1025089">two</span> <span m="1025790">disjoint</span> <span m="1026560">trees</span> <span m="1026900">that</span> <span m="1027020">cannot</span> <span m="1027280">even</span> <span m="1027480">communicate</span> <span m="1028089">to</span> <span m="1028190">one</span> <span m="1028339">another.</span> <span m="1029490">So</span> <span m="1030980">this</span> <span m="1031170">brings</span> <span m="1031500">us</span> <span m="1031760">to</span> <span m="1033819">the</span> <span m="1033970">idea</span> <span m="1034240">of</span> <span m="1034349">congestion.</span> <span m="1035349">And</span> <span m="1037030">in</span> <span m="1037220">order</span> <span m="1037369">to</span> <span m="1038109">define it</span> <span m="1038560">better,</span> <span m="1039190">you will</span> <span m="1039440">need</span> <span m="1039619">a</span> <span m="1039660">few</span> <span m="1039839">definitions.</span> </p>
<p><span m="1044210">So</span> <span m="1046810">to</span> <span m="1046880">start,</span> <span m="1047800">we</span> <span m="1048000">will</span> <span m="1048510">define</span> <span m="1048830">a</span> <span m="1048890">permutation</span> <span m="1051640">and</span> <span m="1051780">we will</span> <span m="1052080">use</span> <span m="1052430">this</span> <span m="1053160">to</span> <span m="1057080">stipulate</span> <span m="1057720">the</span> <span m="1057830">requirement</span> <span m="1058670">that</span> <span m="1058850">we</span> <span m="1059000">want</span> <span m="1059350">on</span> <span m="1059710">how</span> <span m="1060800">inputs</span> <span m="1061290">and</span> <span m="1061420">outputs</span> <span m="1062130">are</span> <span m="1062760">related</span> <span m="1063130">to</span> <span m="1063280">another,</span> <span m="1063630">which</span> <span m="1063960">input</span> <span m="1065260">needs</span> <span m="1065600">to</span> <span m="1065930">communicate</span> <span m="1066500">to</span> <span m="1066660">which</span> <span m="1066960">output.</span> <span m="1068470">So</span> <span m="1068590">permutation</span> <span m="1069790">is</span> <span m="1070070">a</span> <span m="1070230">function</span> <span m="1074580">pi</span> <span m="1076650">from</span> <span m="1076980">the</span> <span m="1077090">set</span> <span m="1077640">0</span> <span m="1077940">to</span> <span m="1078170">n</span> <span m="1078350">minus</span> <span m="1078740">1</span> <span m="1080010">to</span> <span m="1080600">the</span> <span m="1080710">same</span> <span m="1081050">set.</span> <span m="1083750">And</span> <span m="1086430">it</span> <span m="1086630">is</span> <span m="1086740">such</span> <span m="1087100">that</span> <span m="1090020">no</span> <span m="1090280">two</span> <span m="1090460">numbers</span> <span m="1095880">are</span> <span m="1096040">mapped</span> <span m="1096310">to</span> <span m="1098370">more</span> <span m="1098580">than</span> <span m="1098740">once.</span> <span m="1101210">So</span> <span m="1101570">no</span> <span m="1101780">two</span> <span m="1101960">numbers</span> <span m="1102500">are</span> <span m="1102700">mapped</span> <span m="1103010">to</span> <span m="1105130">the</span> <span m="1105280">same</span> <span m="1105550">value.</span> <span m="1111790">So</span> <span m="1112200">what</span> <span m="1112380">we</span> <span m="1112500">really</span> <span m="1112760">want--</span> <span m="1113570">to</span> <span m="1113690">put</span> <span m="1113810">it</span> <span m="1113910">in</span> <span m="1113970">mathematics,</span> <span m="1114760">we</span> <span m="1115550">want</span> <span m="1115880">that</span> <span m="1116580">pi</span> <span m="1116730">of</span> <span m="1116970">i</span> <span m="1117540">is</span> <span m="1117800">only</span> <span m="1118260">equal</span> <span m="1119330">to</span> <span m="1119550">pi</span> <span m="1119840">of</span> <span m="1119980">j</span> <span m="1121600">if</span> <span m="1121790">and</span> <span m="1121910">only</span> <span m="1122240">if</span> <span m="1123600">i</span> <span m="1124030">is</span> <span m="1124280">equal</span> <span m="1124580">to</span> <span m="1124690">j.</span> </p>
<p><span m="1126560">So</span> <span m="1126720">let's</span> <span m="1126920">have</span> <span m="1127570">an</span> <span m="1127740">example</span> <span m="1129540">to</span> <span m="1130020">plug</span> <span m="1130330">into</span> <span m="1130560">that</span> <span m="1130690">picture</span> <span m="1131070">over</span> <span m="1131290">there.</span> <span m="1132480">So</span> <span m="1132830">a</span> <span m="1132890">first</span> <span m="1133180">example</span> <span m="1133690">could</span> <span m="1133870">be</span> <span m="1139000">pi of</span> <span m="1139440">i</span> <span m="1139820">equals,</span> <span m="1140310">say,</span> <span m="1141020">n</span> <span m="1141230">minus</span> <span m="1141630">1</span> <span m="1142600">minus</span> <span m="1143020">i.</span> <span m="1143230">This</span> <span m="1143420">is</span> <span m="1143540">a</span> <span m="1143610">proper</span> <span m="1144230">permutation.</span> <span m="1145880">No</span> <span m="1146090">two</span> <span m="1146250">numbers</span> <span m="1146570">map</span> <span m="1146800">to</span> <span m="1146910">the</span> <span m="1147000">same</span> <span m="1147240">value.</span> <span m="1148170">Another</span> <span m="1148490">one</span> <span m="1148720">could</span> <span m="1148960">be</span> <span m="1149650">the</span> <span m="1149840">identity</span> <span m="1150420">permutation,</span> <span m="1151240">like</span> <span m="1151660">you</span> <span m="1151820">map</span> <span m="1153020">i</span> <span m="1153260">to</span> <span m="1153460">the</span> <span m="1153540">same</span> <span m="1154180">i.</span> <span m="1155730">So</span> <span m="1155850">that's</span> <span m="1156050">another</span> <span m="1156300">example.</span> </p>
<p><span m="1157400">Now,</span> <span m="1157530">how</span> <span m="1157680">do</span> <span m="1157840">we</span> <span m="1158060">use</span> <span m="1158350">permutations</span> <span m="1159550">to</span> <span m="1160460">go</span> <span m="1160640">towards</span> <span m="1160970">the</span> <span m="1161070">idea</span> <span m="1161540">of</span> <span m="1161660">congestion?</span> <span m="1162310">So</span> <span m="1163050">permutation</span> <span m="1164030">can</span> <span m="1164210">be</span> <span m="1164330">used</span> <span m="1164570">to</span> <span m="1164640">formulate</span> <span m="1165230">the</span> <span m="1165980">permutation</span> <span m="1166610">routing</span> <span m="1166970">problem.</span> <span m="1169860">And</span> <span m="1170030">the</span> <span m="1170120">permutation</span> <span m="1171270">routing</span> <span m="1171690">problem</span> <span m="1174960">is</span> <span m="1175570">defined</span> <span m="1175970">like</span> <span m="1176140">this.</span> <span m="1182170">It's</span> <span m="1182370">defined</span> <span m="1182960">as</span> <span m="1183800">follows.</span> <span m="1184490">What</span> <span m="1184660">we</span> <span m="1184860">want</span> <span m="1185130">is</span> <span m="1185250">that</span> <span m="1185550">for</span> <span m="1185850">each</span> <span m="1188020">i,</span> <span m="1189060">we</span> <span m="1189210">want</span> <span m="1189430">to</span> <span m="1189570">direct</span> <span m="1192730">the</span> <span m="1192850">packet</span> <span m="1196670">at</span> <span m="1196960">input</span> <span m="1198860">i</span> <span m="1200170">to</span> <span m="1201460">output</span> <span m="1203010">pi</span> <span m="1203250">of</span> <span m="1203320">i.</span> <span m="1204940">So</span> <span m="1205110">you</span> <span m="1205220">want</span> <span m="1205340">to</span> <span m="1205410">do</span> <span m="1205500">that</span> <span m="1205690">for</span> <span m="1205870">all</span> <span m="1206420">i.</span> </p>
<p><span m="1207166">So</span> <span m="1207540">let's</span> <span m="1207780">have</span> <span m="1208160">a</span> <span m="1208220">look</span> <span m="1208530">at</span> <span m="1209910">this</span> <span m="1210080">particular</span> <span m="1210460">example,</span> <span m="1211080">where</span> <span m="1211240">we</span> <span m="1211320">look</span> <span m="1211510">at</span> <span m="1211660">identity</span> <span m="1212120">permutation.</span> <span m="1213700">So</span> <span m="1213890">if</span> <span m="1213980">you</span> <span m="1214050">do</span> <span m="1214200">that,</span> <span m="1214660">we</span> <span m="1215240">can</span> <span m="1216420">easily</span> <span m="1218200">route</span> <span m="1218515">this,</span> <span m="1218830">right?</span> <span m="1219150">So</span> <span m="1219410">I</span> <span m="1219480">want</span> <span m="1219700">to</span> <span m="1220040">send</span> <span m="1220290">a</span> <span m="1220390">packet</span> <span m="1220660">from</span> <span m="1220930">input</span> <span m="1221230">zero</span> <span m="1221870">to</span> <span m="1222060">output</span> <span m="1222710">zero.</span> <span m="1223610">So</span> <span m="1223740">I</span> <span m="1223780">can</span> <span m="1223930">simply</span> <span m="1225710">go</span> <span m="1225940">into</span> <span m="1226260">this</span> <span m="1226490">direction.</span> <span m="1227040">I</span> <span m="1227120">just</span> <span m="1227410">go</span> <span m="1227580">towards</span> <span m="1227870">this</span> <span m="1228140">switch</span> <span m="1228455">and it</span> <span m="1228770">gets</span> <span m="1229290">routed</span> <span m="1229650">back</span> <span m="1229890">to</span> <span m="1230130">this</span> <span m="1230430">one.</span> <span m="1231580">I</span> <span m="1231740">can</span> <span m="1233360">go</span> <span m="1233550">like</span> <span m="1233760">this</span> <span m="1234920">and</span> <span m="1235090">this</span> <span m="1235280">one</span> <span m="1236030">can</span> <span m="1236210">go</span> <span m="1236430">like</span> <span m="1236680">this</span> <span m="1238170">and</span> <span m="1238320">this</span> <span m="1238500">one</span> <span m="1238620">goes</span> <span m="1238930">like</span> <span m="1239130">that.</span> </p>
<p><span m="1240630">Now, if</span> <span m="1240740">you</span> <span m="1240840">look</span> <span m="1241010">at</span> <span m="1241060">the</span> <span m="1241200">other</span> <span m="1241420">permutation,</span> <span m="1242530">the</span> <span m="1242630">picture</span> <span m="1242970">looks</span> <span m="1243220">very</span> <span m="1243430">different.</span> <span m="1245340">Now,</span> <span m="1246670">we</span> <span m="1246820">want</span> <span m="1247000">to</span> <span m="1247180">route</span> <span m="1248000">input</span> <span m="1248540">zero</span> <span m="1249640">to</span> <span m="1251370">output</span> <span m="1251850">three.</span> <span m="1253200">In order</span> <span m="1253430">to</span> <span m="1253490">do</span> <span m="1253640">this,</span> <span m="1254570">I</span> <span m="1254730">will</span> <span m="1255090">actually</span> <span m="1255450">need</span> <span m="1255680">to</span> <span m="1255800">go</span> <span m="1256420">all</span> <span m="1256670">the</span> <span m="1256780">way</span> <span m="1257760">through</span> <span m="1258150">here</span> <span m="1259740">and</span> <span m="1259870">then</span> <span m="1260040">all</span> <span m="1260210">the</span> <span m="1260300">way</span> <span m="1260480">down</span> <span m="1264660">to</span> <span m="1264780">this</span> <span m="1264950">particular</span> <span m="1265330">output.</span> <span m="1267420">And</span> <span m="1267570">now,</span> <span m="1267690">the</span> <span m="1267780">picture</span> <span m="1268100">gets</span> <span m="1268580">into</span> <span m="1268860">a</span> <span m="1268920">big</span> <span m="1269120">mess</span> <span m="1269550">because</span> <span m="1271210">for</span> <span m="1271400">input</span> <span m="1271710">one,</span> <span m="1272410">we</span> <span m="1272520">have</span> <span m="1272690">to</span> <span m="1272800">go</span> <span m="1273030">to</span> <span m="1273330">output</span> <span m="1275650">two.</span> <span m="1276600">So</span> <span m="1276900">for</span> <span m="1277010">input</span> <span m="1277280">one,</span> <span m="1278010">well,</span> <span m="1278190">we</span> <span m="1278320">go</span> <span m="1278560">all</span> <span m="1278930">the</span> <span m="1279020">way</span> <span m="1279230">like</span> <span m="1279500">this,</span> <span m="1280500">we</span> <span m="1280630">again</span> <span m="1280960">go</span> <span m="1281180">through</span> <span m="1281940">the</span> <span m="1282040">root,</span> <span m="1282870">and</span> <span m="1283050">then</span> <span m="1283210">we</span> <span m="1283320">go</span> <span m="1283530">down</span> <span m="1286000">to</span> <span m="1286170">this</span> <span m="1286350">particular</span> <span m="1286760">output.</span> </p>
<p><span m="1288740">And</span> <span m="1288920">as</span> <span m="1288990">you</span> <span m="1289130">can</span> <span m="1289310">see,</span> <span m="1289940">for</span> <span m="1291100">input</span> <span m="1291370">two,</span> <span m="1292020">well,</span> <span m="1292620">we</span> <span m="1292720">need</span> <span m="1292850">to</span> <span m="1292950">connect</span> <span m="1293400">to</span> <span m="1293980">output</span> <span m="1294380">one.</span> <span m="1294710">So</span> <span m="1294970">again,</span> <span m="1295230">we</span> <span m="1295330">go</span> <span m="1295580">all</span> <span m="1295990">the</span> <span m="1296080">way</span> <span m="1296320">up</span> <span m="1296980">and we</span> <span m="1297290">go</span> <span m="1297500">all</span> <span m="1297700">the</span> <span m="1297790">way</span> <span m="1297950">down.</span> <span m="1298910">And</span> <span m="1299090">for</span> <span m="1299230">this</span> <span m="1299500">one,</span> <span m="1300500">we</span> <span m="1300630">will</span> <span m="1300810">again</span> <span m="1301070">go</span> <span m="1301270">all</span> <span m="1301520">the</span> <span m="1301590">way</span> <span m="1301800">up</span> <span m="1302110">and</span> <span m="1302350">all</span> <span m="1302560">the</span> <span m="1302660">way</span> <span m="1302810">down</span> <span m="1303150">to</span> <span m="1303390">input</span> <span m="1304100">zero.</span> <span m="1304520">So</span> <span m="1304680">now,</span> <span m="1304830">you</span> <span m="1304940">can</span> <span m="1305100">see</span> <span m="1305850">that</span> <span m="1306040">this</span> <span m="1306250">particular</span> <span m="1307500">switch</span> <span m="1307980">over</span> <span m="1308300">here</span> <span m="1308950">has</span> <span m="1309170">to</span> <span m="1309310">serve</span> <span m="1310190">packets</span> <span m="1310870">from</span> <span m="1312240">all</span> <span m="1313980">the</span> <span m="1314190">inputs.</span> <span m="1315110">All</span> <span m="1315330">the</span> <span m="1315430">four</span> <span m="1316250">packets</span> <span m="1316830">have</span> <span m="1317100">to</span> <span m="1317240">travel</span> <span m="1317650">through</span> <span m="1317920">this</span> <span m="1318260">particular</span> <span m="1318730">node</span> <span m="1319000">here.</span> <span m="1320160">So</span> <span m="1320330">this</span> <span m="1320550">leads</span> <span m="1320810">us</span> <span m="1320980">to</span> <span m="1321730">the</span> <span m="1322010">following</span> <span m="1322400">definition</span> <span m="1323490">of</span> <span m="1323700">congestion.</span> </p>
<p><span m="1325310">So</span> <span m="1325540">the</span> <span m="1325640">congestion--</span> <span m="1332505">oh,</span> <span m="1332970">before</span> <span m="1334390">we</span> <span m="1334490">continue,</span> <span m="1334940">let</span> <span m="1335170">me</span> <span m="1335350">first</span> <span m="1336110">define</span> <span m="1336960">a</span> <span m="1337150">path.</span> <span m="1338240">So</span> <span m="1338490">for</span> <span m="1338650">i,</span> <span m="1338960">we</span> <span m="1339270">direct</span> <span m="1339570">a</span> <span m="1339660">packet</span> <span m="1340290">at</span> <span m="1340810">input</span> <span m="1341170">i</span> <span m="1341480">to</span> <span m="1341650">output</span> <span m="1342080">pi of</span> <span m="1342320">i.</span> <span m="1343300">And</span> <span m="1343900">the</span> <span m="1344000">path</span> <span m="1344800">that</span> <span m="1345090">corresponds</span> <span m="1345770">to</span> <span m="1345890">this</span> <span m="1346220">route</span> <span m="1347010">is</span> <span m="1347220">actually</span> <span m="1347580">denoted</span> <span m="1350840">as</span> <span m="1351020">follows.</span> <span m="1351400">So</span> <span m="1351490">the</span> <span m="1351610">path</span> <span m="1354890">taken</span> <span m="1356880">is</span> <span m="1358470">denoted</span> <span m="1362110">by</span> <span m="1362990">P</span> <span m="1364702">i</span> <span m="1365820">pi</span> <span m="1366620">i.</span> <span m="1368260">So</span> <span m="1368670">now,</span> <span m="1368830">we</span> <span m="1368890">can</span> <span m="1369120">define</span> <span m="1369870">the</span> <span m="1369960">congestion</span> <span m="1370740">of</span> <span m="1370930">a</span> <span m="1371000">set</span> <span m="1371260">of</span> <span m="1371400">such</span> <span m="1371710">paths.</span> <span m="1372170">So</span> <span m="1372320">the</span> <span m="1372400">congestion</span> <span m="1374010">of</span> <span m="1374860">the</span> <span m="1375020">path</span> <span m="1376710">corresponding</span> <span m="1377740">to</span> <span m="1379410">P</span> <span m="1380140">zero</span> <span m="1380750">to</span> <span m="1381570">P</span> <span m="1381930">pi</span> <span m="1382290">zero</span> <span m="1383090">and</span> <span m="1383290">so</span> <span m="1383480">on</span> <span m="1383850">and we go all</span> <span m="1384020">the</span> <span m="1384110">way</span> <span m="1384300">up</span> <span m="1384480">to</span> <span m="1385840">the</span> <span m="1385940">n</span> <span m="1386130">minus</span> <span m="1386480">1</span> <span m="1386940">input</span> <span m="1387670">that</span> <span m="1387810">needs</span> <span m="1388040">to</span> <span m="1388410">be</span> <span m="1388590">mapped</span> <span m="1388960">to</span> <span m="1389060">pi</span> <span m="1389500">of</span> <span m="1389690">n</span> <span m="1390010">minus</span> <span m="1390370">1.</span> </p>
<p><span m="1391680">So</span> <span m="1391880">the</span> <span m="1391980">congestion</span> <span m="1393290">is</span> <span m="1393590">now</span> <span m="1393770">defined</span> <span m="1395430">as</span> <span m="1403210">the</span> <span m="1403350">largest</span> <span m="1403910">number</span> <span m="1407530">of</span> <span m="1407830">paths</span> <span m="1412040">that</span> <span m="1412310">pass</span> <span m="1413190">through</span> <span m="1421500">a</span> <span m="1421600">single</span> <span m="1421920">switch.</span> <span m="1423650">So</span> <span m="1423780">in</span> <span m="1423900">our</span> <span m="1424150">example,</span> <span m="1425340">we</span> <span m="1426980">saw</span> <span m="1427470">that</span> <span m="1428090">in</span> <span m="1428730">the</span> <span m="1428890">case</span> <span m="1429190">of</span> <span m="1429310">the</span> <span m="1429400">blue</span> <span m="1429940">arrows</span> <span m="1430330">here</span> <span m="1431640">for the</span> <span m="1431880">identity</span> <span m="1432340">permutation,</span> <span m="1433520">well,</span> <span m="1435140">this</span> <span m="1435380">switch</span> <span m="1435610">only</span> <span m="1436130">needs</span> <span m="1436410">to</span> <span m="1436640">transmit</span> <span m="1437080">one</span> <span m="1437270">packet</span> <span m="1438320">and</span> <span m="1438700">all those</span> <span m="1439120">actually</span> <span m="1439500">zero</span> <span m="1439830">packets.</span> <span m="1440550">So</span> <span m="1441350">actually,</span> <span m="1441630">the</span> <span m="1441770">congestion</span> <span m="1442500">here</span> <span m="1443830">is</span> <span m="1444350">equal</span> <span m="1449670">to</span> <span m="1450080">1.</span> <span m="1451080">And</span> <span m="1451520">for</span> <span m="1451830">this</span> <span m="1452050">particular</span> <span m="1452440">permutation,</span> <span m="1454370">well,</span> <span m="1454670">we</span> <span m="1454760">had</span> <span m="1455000">to</span> <span m="1455670">direct</span> <span m="1456190">all</span> <span m="1456440">the</span> <span m="1456550">packets</span> <span m="1457210">through</span> <span m="1457490">the</span> <span m="1457620">root</span> <span m="1458260">and</span> <span m="1458450">it's</span> <span m="1458610">the</span> <span m="1459880">most</span> <span m="1460170">accessed</span> <span m="1460660">switch.</span> <span m="1461470">And</span> <span m="1461650">that</span> <span m="1461720">switch</span> <span m="1462560">has</span> <span m="1462780">congestion</span> <span m="1463360">four,</span> <span m="1463900">right?</span> <span m="1464830">So</span> <span m="1467520">the</span> <span m="1467620">congestion</span> <span m="1468280">over</span> <span m="1468560">here</span> <span m="1470950">is</span> <span m="1471110">equal</span> <span m="1471310">to</span> <span m="1471420">4.</span> </p>
<p><span m="1474520">Now,</span> <span m="1474650">this</span> <span m="1474790">does not</span> <span m="1475220">look</span> <span m="1475390">so</span> <span m="1475560">good</span> <span m="1476460">because</span> <span m="1477670">for</span> <span m="1477900">a</span> <span m="1477930">binary</span> <span m="1478430">tree,</span> <span m="1479080">we</span> <span m="1479220">always</span> <span m="1479760">have</span> <span m="1479980">this</span> <span m="1480200">vulnerable</span> <span m="1480790">root</span> <span m="1481820">that is</span> <span m="1482010">right</span> <span m="1482220">here</span> <span m="1482390">in</span> <span m="1482460">the</span> <span m="1482530">center</span> <span m="1482940">connecting</span> <span m="1484620">the</span> <span m="1484710">left</span> <span m="1485040">side</span> <span m="1485380">to</span> <span m="1485480">the</span> <span m="1485610">right</span> <span m="1485890">side.</span> <span m="1486420">So</span> <span m="1486520">we</span> <span m="1486620">can</span> <span m="1486650">always</span> <span m="1486990">find</span> <span m="1487170">a</span> <span m="1487220">permutation--</span> <span m="1487910">actually, this</span> <span m="1488210">permutation</span> <span m="1488940">over</span> <span m="1489200">here--</span> <span m="1490170">that</span> <span m="1490350">leads</span> <span m="1490680">to</span> <span m="1490830">this</span> <span m="1491470">worst-case</span> <span m="1492210">congestion.</span> <span m="1494110">So</span> <span m="1495780">what</span> <span m="1495920">we're</span> <span m="1496070">interested</span> <span m="1496690">in</span> <span m="1497030">is</span> <span m="1498150">the</span> <span m="1498260">maximum</span> <span m="1498820">congestion,</span> <span m="1499470">which</span> <span m="1499670">is</span> <span m="1499740">sort</span> <span m="1499980">of</span> <span m="1500090">the</span> <span m="1500190">worst-case</span> <span m="1500750">scenario.</span> <span m="1502520">And we'll</span> <span m="1502910">define</span> <span m="1503360">it</span> <span m="1506890">as</span> <span m="1507110">follows.</span> </p>
<p><span m="1509150">The</span> <span m="1509230">maximum</span> <span m="1509640">congestion</span> <span m="1511070">is</span> <span m="1511300">actually</span> <span m="1511720">equal</span> <span m="1511990">to</span> <span m="1512120">the</span> <span m="1512220">maximum</span> <span m="1514610">over</span> <span m="1514950">all</span> <span m="1515250">permutations</span> <span m="1516090">pi.</span> <span m="1516450">So</span> <span m="1516580">this</span> <span m="1516740">is</span> <span m="1516900">kind</span> <span m="1517150">of</span> <span m="1517430">the</span> <span m="1517790">worst-case</span> <span m="1519060">routing</span> <span m="1519460">problem</span> <span m="1519840">that</span> <span m="1519990">I</span> <span m="1520080">can</span> <span m="1520250">imagine</span> <span m="1521080">and</span> <span m="1521260">it</span> <span m="1521410">may</span> <span m="1521710">occur</span> <span m="1522050">in</span> <span m="1522140">practice.</span> <span m="1523180">So</span> <span m="1523480">in</span> <span m="1523600">the</span> <span m="1523730">worst</span> <span m="1524080">case,</span> <span m="1526210">how</span> <span m="1526510">can</span> <span m="1526760">I</span> <span m="1527390">solve</span> <span m="1527900">it</span> <span m="1528110">the</span> <span m="1528200">best?</span> <span m="1528950">So</span> <span m="1529600">I</span> <span m="1529740">want</span> <span m="1529920">to</span> <span m="1530030">find</span> <span m="1530410">the</span> <span m="1530500">minimum</span> <span m="1532360">of</span> <span m="1532630">the</span> <span m="1532770">congestion</span> <span m="1537440">of</span> <span m="1537735">a</span> <span m="1538030">path</span> <span m="1545810">over</span> <span m="1546100">here</span> <span m="1547510">and</span> <span m="1548400">the</span> <span m="1548620">minimum</span> <span m="1549010">is</span> <span m="1549260">over</span> <span m="1549580">these</span> <span m="1549730">types of</span> <span m="1550160">paths.</span> <span m="1550750">So</span> <span m="1551370">actually,</span> <span m="1552190">this</span> <span m="1552400">is</span> <span m="1553450">our</span> <span m="1553770">solution</span> <span m="1554840">to</span> <span m="1555300">this</span> <span m="1555600">routing</span> <span m="1555970">problem.</span> <span m="1556480">We</span> <span m="1556580">want</span> <span m="1556740">to</span> <span m="1556820">find</span> <span m="1557060">the</span> <span m="1557140">best</span> <span m="1557540">kind</span> <span m="1557740">of</span> <span m="1557800">solution</span> <span m="1558740">for this</span> <span m="1559180">worst-case</span> <span m="1560020">scenario--</span> <span m="1561760">so</span> <span m="1565950">the</span> <span m="1565990">minimum</span> <span m="1566410">over</span> <span m="1566760">all</span> <span m="1566960">solutions</span> <span m="1568020">for</span> <span m="1569040">these</span> <span m="1569410">paths</span> </p>
<p><span m="1578750">So</span> <span m="1580180">well,</span> <span m="1581940">for</span> <span m="1582960">this</span> <span m="1583160">particular</span> <span m="1583640">tree</span> <span m="1583970">structure,</span> <span m="1585420">this</span> <span m="1586920">permutation</span> <span m="1588320">is</span> <span m="1588520">really</span> <span m="1588890">the</span> <span m="1589160">worst-case</span> <span m="1589680">scenario</span> <span m="1590040">that</span> <span m="1590200">you</span> <span m="1590300">can</span> <span m="1590460">have</span> <span m="1590760">because</span> <span m="1591540">every</span> <span m="1591980">packet</span> <span m="1592360">needs</span> <span m="1592520">to</span> <span m="1592630">be</span> <span m="1592790">routed</span> <span m="1593150">through</span> <span m="1593320">the</span> <span m="1593440">center</span> <span m="1594680">over</span> <span m="1594980">here.</span> <span m="1595700">And</span> <span m="1595750">it</span> <span m="1595870">means</span> <span m="1596150">that</span> <span m="1596370">our</span> <span m="1597020">maximum</span> <span m="1597580">congestion</span> <span m="1598520">for</span> <span m="1598840">an</span> <span m="1599040">arbitrary</span> <span m="1599520">tree</span> <span m="1600260">is</span> <span m="1600440">actually</span> <span m="1600790">equal</span> <span m="1601810">to</span> <span m="1602010">n.</span> <span m="1604960">So</span> <span m="1607140">that</span> <span m="1608670">looks</span> <span m="1608880">really</span> <span m="1609090">bad,</span> <span m="1609360">actually.</span> <span m="1609950">So</span> <span m="1610130">we</span> <span m="1610230">don't</span> <span m="1610410">like</span> <span m="1610640">this</span> <span m="1612240">at</span> <span m="1612390">all.</span> </p>
<p><span m="1613650">So</span> <span m="1614360">let's</span> <span m="1614650">find</span> <span m="1614910">out</span> <span m="1615120">where</span> <span m="1615230">we</span> <span m="1615400">can</span> <span m="1615550">do</span> <span m="1615690">a</span> <span m="1615720">little</span> <span m="1615890">bit</span> <span m="1616060">better</span> <span m="1616960">and</span> <span m="1617630">we</span> <span m="1617920">come</span> <span m="1618700">to</span> <span m="1618850">look</span> <span m="1619060">at</span> <span m="1619290">the</span> <span m="1620000">two-dimensional</span> <span m="1620660">array</span> <span m="1621350">and</span> <span m="1621550">see</span> <span m="1623270">what</span> <span m="1623430">that</span> <span m="1623560">would</span> <span m="1624080">lead</span> <span m="1624290">up</span> <span m="1624420">to.</span> <span m="1625770">And</span> <span m="1627730">its</span> <span m="1627990">structure</span> <span m="1628780">is</span> <span m="1629200">as</span> <span m="1629370">follows.</span> <span m="1633920">We</span> <span m="1634020">essentially</span> <span m="1634470">have</span> <span m="1634860">inputs</span> <span m="1635780">on</span> <span m="1636830">the</span> <span m="1637030">left</span> <span m="1637950">and</span> <span m="1639650">the</span> <span m="1639790">outputs</span> <span m="1640200">are</span> <span m="1640310">on</span> <span m="1640380">the</span> <span m="1640460">bottom</span> <span m="1640910">and</span> <span m="1641010">they</span> <span m="1641110">are</span> <span m="1641320">in</span> <span m="1641510">a</span> <span m="1641550">grid</span> <span m="1641840">structure.</span> <span m="1643280">So</span> <span m="1643520">we</span> <span m="1643640">have</span> <span m="1643910">input</span> <span m="1644280">zero,</span> <span m="1645660">input</span> <span m="1646050">one,</span> <span m="1647450">input</span> <span m="1647860">two,</span> <span m="1649100">input</span> <span m="1649480">three.</span> <span m="1650700">They</span> <span m="1650910">all</span> <span m="1651140">connect</span> <span m="1651570">to</span> <span m="1651690">their</span> <span m="1651940">terminals.</span> <span m="1655660">We</span> <span m="1655810">have</span> <span m="1655990">switches,</span> <span m="1660380">four</span> <span m="1661080">of</span> <span m="1661290">those,</span> <span m="1662640">and</span> <span m="1662910">they</span> <span m="1663090">are</span> <span m="1663250">all</span> <span m="1664810">connected</span> <span m="1665450">in</span> <span m="1665640">this</span> <span m="1665870">grid.</span> </p>
<p><span m="1669660">And</span> <span m="1671170">at</span> <span m="1671320">the</span> <span m="1671480">very</span> <span m="1671660">bottom,</span> <span m="1672270">we will</span> <span m="1672520">have</span> <span m="1672690">the</span> <span m="1672810">outputs,</span> <span m="1675100">the</span> <span m="1675240">output</span> <span m="1675570">terminals.</span> <span m="1688580">So</span> <span m="1688800">this</span> <span m="1689000">is</span> <span m="1689180">output</span> <span m="1690880">zero</span> <span m="1692240">and</span> <span m="1692400">here,</span> <span m="1692630">we</span> <span m="1692730">will</span> <span m="1692830">have</span> <span m="1693140">output</span> <span m="1693670">one,</span> <span m="1694940">output</span> <span m="1695440">two,</span> <span m="1696100">and</span> <span m="1696350">output</span> <span m="1697320">three.</span> <span m="1703000">So</span> <span m="1703210">notice</span> <span m="1703500">that</span> <span m="1703600">my</span> <span m="1703740">circle</span> <span m="1704200">start</span> <span m="1704525">to</span> <span m="1704850">resemble</span> <span m="1705150">my</span> <span m="1705270">squares,</span> <span m="1707000">but</span> <span m="1707950">these</span> <span m="1708160">are all the</span> <span m="1708580">switches</span> <span m="1708940">right</span> <span m="1709130">here</span> <span m="1709320">in</span> <span m="1709390">the</span> <span m="1709470">center.</span> </p>
<p><span m="1710630">So</span> <span m="1711370">how</span> <span m="1711520">does</span> <span m="1711720">this</span> <span m="1711920">work?</span> <span m="1712650">Well,</span> <span m="1715350">do</span> <span m="1715420">we</span> <span m="1715520">have</span> <span m="1715690">a</span> <span m="1715780">better</span> <span m="1716380">parameter?</span> <span m="1717300">So</span> <span m="1717470">let's</span> <span m="1717860">look</span> <span m="1718070">at</span> <span m="1718130">it</span> <span m="1718240">together.</span> <span m="1719280">So</span> <span m="1720480">we</span> <span m="1720580">need</span> <span m="1720750">to</span> <span m="1720850">first</span> <span m="1721140">of</span> <span m="1721190">all</span> <span m="1721530">figure</span> <span m="1721890">out</span> <span m="1722160">what</span> <span m="1722250">the</span> <span m="1722350">diameter</span> <span m="1722780">is.</span> <span m="1723390">So</span> <span m="1723530">what's</span> <span m="1723710">the</span> <span m="1723770">diameter</span> <span m="1724280">of</span> <span m="1724400">this</span> <span m="1724550">particular</span> <span m="1725640">network?</span> <span m="1728310">So</span> <span m="1730240">what's</span> <span m="1730490">the</span> <span m="1730560">shortest</span> <span m="1731170">path</span> <span m="1731720">between</span> <span m="1731960">the</span> <span m="1732060">furthest</span> <span m="1734050">input</span> <span m="1734750">and</span> <span m="1735210">output?</span> <span m="1736860">So</span> <span m="1738020">if</span> <span m="1738210">you</span> <span m="1738340">look</span> <span m="1738530">at</span> <span m="1738630">that,</span> <span m="1739040">we</span> <span m="1739310">can</span> <span m="1739530">see</span> <span m="1739880">that</span> <span m="1740590">if</span> <span m="1740820">I</span> <span m="1740940">go</span> <span m="1741570">all</span> <span m="1741790">the</span> <span m="1741840">way</span> <span m="1741990">from</span> <span m="1742210">here</span> <span m="1742530">and</span> <span m="1742650">I</span> <span m="1742690">go all</span> <span m="1743000">the</span> <span m="1743070">way</span> <span m="1743200">down</span> <span m="1743460">to</span> <span m="1743580">this</span> <span m="1743800">corner,</span> <span m="1745050">that</span> <span m="1745230">looks</span> <span m="1745470">like</span> <span m="1745700">the</span> <span m="1745800">largest</span> <span m="1746880">path</span> <span m="1747480">and</span> <span m="1747710">I</span> <span m="1747770">need</span> <span m="1748000">to</span> <span m="1748490">cross</span> <span m="1748860">all</span> <span m="1749100">these</span> <span m="1749820">wires.</span> <span m="1751530">And</span> <span m="1751770">in</span> <span m="1751840">general,</span> <span m="1752610">for</span> <span m="1752880">any</span> <span m="1753180">n,</span> <span m="1754780">we</span> <span m="1754950">will</span> <span m="1755160">have</span> <span m="1756100">that</span> <span m="1757110">the</span> <span m="1757250">diameter</span> <span m="1757950">is</span> <span m="1758320">2</span> <span m="1758540">times</span> <span m="1759710">n.</span> </p>
<p><span m="1760760">Now,</span> <span m="1760870">what</span> <span m="1761020">about</span> <span m="1761260">the</span> <span m="1761340">switch</span> <span m="1761960">size?</span> <span m="1763760">It</span> <span m="1763850">looks</span> <span m="1764070">a</span> <span m="1764100">little</span> <span m="1764280">bit</span> <span m="1764420">smaller,</span> <span m="1764880">right?</span> <span m="1765170">Because</span> <span m="1765530">over</span> <span m="1765800">here,</span> <span m="1766030">we</span> <span m="1766140">had</span> <span m="1766340">three</span> <span m="1766700">inputs</span> <span m="1767110">coming</span> <span m="1767410">in</span> <span m="1767530">and</span> <span m="1767600">three</span> <span m="1767840">outputs</span> <span m="1768220">coming</span> <span m="1768520">out</span> <span m="1769850">but</span> <span m="1770010">over</span> <span m="1770320">here,</span> <span m="1771140">we see that</span> <span m="1771620">every</span> <span m="1771900">single</span> <span m="1772230">switch</span> <span m="1772620">is</span> <span m="1772760">only</span> <span m="1773030">two</span> <span m="1773270">inputs</span> <span m="1773560">and</span> <span m="1773850">two</span> <span m="1774270">outputs.</span> <span m="1775270">So</span> <span m="1775500">that</span> <span m="1775660">makes the</span> <span m="1775960">size</span> <span m="1777400">two</span> <span m="1777630">times</span> <span m="1778030">two.</span> <span m="1781950">Now,</span> <span m="1782100">the</span> <span m="1782200">number</span> <span m="1782520">of</span> <span m="1782590">switches</span> <span m="1783130">is</span> <span m="1783290">pretty</span> <span m="1783560">bad,</span> <span m="1783890">right,</span> <span m="1784590">because</span> <span m="1784940">we</span> <span m="1785070">have</span> <span m="1785410">n</span> <span m="1785680">squared</span> <span m="1788510">switches.</span> <span m="1789140">So</span> <span m="1789580">that's</span> <span m="1789800">really</span> <span m="1789990">horrible.</span> <span m="1790790">That's</span> <span m="1791480">a</span> <span m="1791550">lot.</span> <span m="1792100">We</span> <span m="1792260">would</span> <span m="1792410">like</span> <span m="1792590">to</span> <span m="1792680">do</span> <span m="1792790">much</span> <span m="1793060">better.</span> </p>
<p><span m="1794330">And</span> <span m="1795070">what</span> <span m="1795220">about</span> <span m="1795430">the</span> <span m="1795520">congestion?</span> <span m="1796270">Do you</span> <span m="1796640">have</span> <span m="1796910">any</span> <span m="1797140">idea</span> <span m="1797930">what</span> <span m="1798120">the</span> <span m="1798200">congestion</span> <span m="1798760">could</span> <span m="1798960">be</span> <span m="1799160">in</span> <span m="1799260">this</span> <span m="1799450">particular</span> <span m="1799920">case?</span> <span m="1800450">We</span> <span m="1800480">will</span> <span m="1800800">prove</span> <span m="1801710">a</span> <span m="1801800">theorem</span> <span m="1802450">on</span> <span m="1802690">that.</span> <span m="1807190">For</span> <span m="1807410">any</span> <span m="1807740">permutation,</span> <span m="1808040">is</span> <span m="1808340">there</span> <span m="1808490">a way</span> <span m="1811180">to</span> <span m="1811510">route</span> <span m="1811990">the</span> <span m="1812140">inputs</span> <span m="1812570">to</span> <span m="1812690">the</span> <span m="1812910">outputs</span> <span m="1813350">in</span> <span m="1813450">such a</span> <span m="1813905">way</span> <span m="1814360">that</span> <span m="1814600">the</span> <span m="1814710">switches</span> <span m="1815060">get</span> <span m="1815980">almost</span> <span m="1816470">not</span> <span m="1816700">congested?</span> <span m="1817790">So</span> <span m="1817830">in</span> <span m="1817920">the</span> <span m="1818000">binary</span> <span m="1818370">tree,</span> <span m="1818710">we</span> <span m="1818820">had</span> <span m="1819030">a</span> <span m="1819070">congestion</span> <span m="1819640">of</span> <span m="1820500">n,</span> <span m="1820740">which</span> <span m="1820920">is</span> <span m="1821050">linear</span> <span m="1821400">in the</span> <span m="1821530">switches.</span> </p>
<p><span m="1823080">But</span> <span m="1823640">over</span> <span m="1823950">here,</span> <span m="1824120">we</span> <span m="1824220">can</span> <span m="1824410">do</span> <span m="1824590">much</span> <span m="1824880">better.</span> <span m="1826780">We</span> <span m="1826890">will</span> <span m="1827050">show</span> <span m="1828840">that</span> <span m="1831440">the</span> <span m="1831530">congestion</span> <span m="1837630">of</span> <span m="1838280">an</span> <span m="1840190">n-input</span> <span m="1843660">array</span> <span m="1845210">is</span> <span m="1845410">actually</span> <span m="1845680">equal</span> <span m="1845970">to</span> <span m="1846110">2.</span> <span m="1847700">So</span> <span m="1847860">that's</span> <span m="1848110">great.</span> <span m="1849590">So</span> <span m="1849800">I'll</span> <span m="1850370">prove</span> <span m="1850470">it in</span> <span m="1850540">a</span> <span m="1850590">moment,</span> <span m="1851050">but</span> <span m="1852060">that</span> <span m="1852230">looks</span> <span m="1852460">really</span> <span m="1852700">fantastic.</span> <span m="1853960">And</span> <span m="1855690">so</span> <span m="1855860">it's</span> <span m="1856110">way</span> <span m="1856370">better</span> <span m="1856940">than</span> <span m="1857430">the</span> <span m="1857510">binary</span> <span m="1857920">tree.</span> </p>
<p><span m="1858930">Now,</span> <span m="1859070">this</span> <span m="1859320">is</span> <span m="1859480">really</span> <span m="1859710">not</span> <span m="1859890">so</span> <span m="1860070">good</span> <span m="1860660">and</span> <span m="1860760">this</span> <span m="1860900">is</span> <span m="1861020">also</span> <span m="1861320">much</span> <span m="1861590">larger,</span> <span m="1862030">but</span> <span m="1862930">still--</span> <span m="1866090">we</span> <span m="1866210">will</span> <span m="1867300">start</span> <span m="1867660">to</span> <span m="1867760">think</span> <span m="1868680">next</span> <span m="1869440">after</span> <span m="1870040">we</span> <span m="1870170">show</span> <span m="1870420">this</span> <span m="1870630">particular</span> <span m="1871040">property</span> <span m="1872380">how</span> <span m="1872590">to</span> <span m="1872730">combine</span> <span m="1873180">these</span> <span m="1873510">two</span> <span m="1874350">and</span> <span m="1874550">see</span> <span m="1874740">how</span> <span m="1874930">we</span> <span m="1875050">can</span> <span m="1875230">come</span> <span m="1875470">up</span> <span m="1875770">with</span> <span m="1876150">another</span> <span m="1876460">network</span> <span m="1877190">that's</span> <span m="1878920">able</span> <span m="1879150">to</span> <span m="1879240">combine</span> <span m="1879860">in</span> <span m="1879980">some</span> <span m="1880240">ways</span> <span m="1880670">these</span> <span m="1880830">two</span> <span m="1880990">properties.</span> <span m="1882130">And</span> <span m="1882230">maybe</span> <span m="1882410">we</span> <span m="1882540">can</span> <span m="1882700">find</span> <span m="1883180">a</span> <span m="1883280">good</span> <span m="1883450">solution</span> <span m="1883900">that</span> <span m="1884080">way.</span> <span m="1885430">It turns</span> <span m="1885750">out</span> <span m="1885900">we</span> <span m="1886060">will</span> <span m="1886100">not</span> <span m="1886410">immediately</span> <span m="1886690">be</span> <span m="1886800">able</span> <span m="1886980">to</span> <span m="1887090">do</span> <span m="1887250">that.</span> <span m="1887440">We</span> <span m="1887800">will</span> <span m="1888040">need</span> <span m="1888180">to</span> <span m="1888320">make</span> <span m="1888500">another</span> <span m="1888780">step</span> <span m="1889310">and</span> <span m="1889530">come</span> <span m="1889720">to</span> <span m="1889820">the</span> <span m="1889960">last</span> <span m="1890940">network.</span> <span m="1891700">It</span> <span m="1891760">really</span> <span m="1891990">has</span> <span m="1892220">good</span> <span m="1893220">parameters.</span> </p>
<p><span m="1894330">So</span> <span m="1894490">what</span> <span m="1894610">about</span> <span m="1894830">the</span> <span m="1894920">theorem?</span> <span m="1896500">So</span> <span m="1896870">if</span> <span m="1896960">you</span> <span m="1897080">prove</span> <span m="1897580">this,</span> <span m="1899130">well,</span> <span m="1899330">how</span> <span m="1899440">do</span> <span m="1899540">we</span> <span m="1899650">start?</span> <span m="1900930">You</span> <span m="1901050">just</span> <span m="1901420">start</span> <span m="1902680">with</span> <span m="1902970">any</span> <span m="1904010">permutation.</span> <span m="1904860">If</span> <span m="1905020">I</span> <span m="1905050">want</span> <span m="1905220">to</span> <span m="1905310">prove</span> <span m="1905530">something</span> <span m="1905880">about the</span> <span m="1906130">congestion,</span> <span m="1906710">it's</span> <span m="1906920">defined</span> <span m="1907380">as</span> <span m="1907540">the</span> <span m="1908050">maximum</span> <span m="1908920">of</span> <span m="1909150">all</span> <span m="1909330">permutations.</span> <span m="1909775">So</span> <span m="1910220">let's</span> <span m="1911380">take</span> <span m="1911600">one</span> <span m="1911760">of</span> <span m="1911880">them</span> <span m="1912510">and</span> <span m="1912680">see</span> <span m="1912830">what</span> <span m="1912990">we</span> <span m="1913080">can</span> <span m="1913230">prove.</span> </p>
<p><span m="1927359">So</span> <span m="1927810">let</span> <span m="1927990">us</span> <span m="1928170">define</span> <span m="1929300">the</span> <span m="1929400">paths</span> <span m="1930010">for</span> <span m="1930160">this</span> <span m="1930350">permutation.</span> <span m="1931305">So what</span> <span m="1931600">we really</span> <span m="1931830">want</span> <span m="1931980">to</span> <span m="1932060">do</span> <span m="1932220">is</span> <span m="1932830">we</span> <span m="1932990">take</span> <span m="1933250">any</span> <span m="1933580">permutation</span> <span m="1934530">and</span> <span m="1934690">we</span> <span m="1934800">want</span> <span m="1935070">to</span> <span m="1935190">find</span> <span m="1935470">a</span> <span m="1935530">really</span> <span m="1935820">good</span> <span m="1936010">solution</span> <span m="1936550">for the</span> <span m="1936780">routing.</span> <span m="1937700">If</span> <span m="1938950">that</span> <span m="1939130">gives</span> <span m="1939350">us</span> <span m="1939510">a</span> <span m="1939550">very</span> <span m="1939790">low</span> <span m="1939960">congestion,</span> <span m="1940255">we</span> <span m="1940550">are</span> <span m="1940640">very</span> <span m="1940970">happy.</span> <span m="1942500">So</span> <span m="1943800">the</span> <span m="1943920">way</span> <span m="1944050">to</span> <span m="1944210">do</span> <span m="1944410">this</span> <span m="1944700">is</span> <span m="1944830">well,</span> <span m="1945125">maybe</span> <span m="1945420">you</span> <span m="1945640">have</span> <span m="1945830">an</span> <span m="1945920">idea</span> <span m="1946230">already.</span> <span m="1948270">So</span> <span m="1948600">how</span> <span m="1948940">would</span> <span m="1949305">I</span> <span m="1949670">route</span> <span m="1950020">this?</span> <span m="1950120">So</span> <span m="1950500">I want to</span> <span m="1950620">connect</span> <span m="1950970">an</span> <span m="1951090">input</span> <span m="1952200">i,</span> <span m="1952630">say,</span> <span m="1952910">1,</span> <span m="1953380">2,</span> <span m="1954900">output</span> <span m="1955260">two,</span> <span m="1955480">for</span> <span m="1955700">example.</span> <span m="1958580">How</span> <span m="1958670">can</span> <span m="1958830">I</span> <span m="1958880">do</span> <span m="1959040">this?</span> <span m="1959680">Any</span> <span m="1959960">suggestions?</span> <span m="1961640">So</span> <span m="1962080">of</span> <span m="1962220">course,</span> <span m="1962540">I</span> <span m="1962640">could</span> <span m="1962720">go</span> <span m="1962940">any</span> <span m="1963240">path,</span> <span m="1964070">but</span> <span m="1964220">somehow,</span> <span m="1964590">I</span> <span m="1964610">want</span> <span m="1964830">to</span> <span m="1965070">have</span> <span m="1965200">some</span> <span m="1965520">uniform</span> <span m="1966060">structure</span> <span m="1967230">that</span> <span m="1968000">hopefully</span> <span m="1968500">helps</span> <span m="1968910">me</span> <span m="1969080">to</span> <span m="1969190">prove</span> <span m="1969610">that</span> <span m="1969690">the</span> <span m="1969800">congestion</span> <span m="1970510">in every</span> <span m="1970900">switch</span> <span m="1971190">is</span> <span m="1971360">very</span> <span m="1971610">small.</span> </p>
<p><span m="1972260">So</span> <span m="1972510">how could</span> <span m="1972910">I</span> <span m="1973170">think</span> <span m="1973430">about</span> <span m="1973750">this?</span> <span m="1976520">Well,</span> <span m="1978330">if</span> <span m="1978610">I</span> <span m="1978680">make</span> <span m="1978940">sure</span> <span m="1979570">that,</span> <span m="1981180">say,</span> <span m="1982080">a</span> <span m="1982220">packet</span> <span m="1982890">that</span> <span m="1983100">goes</span> <span m="1983330">from</span> <span m="1983540">one</span> <span m="1984480">to</span> <span m="1984690">output</span> <span m="1985120">two</span> <span m="1985520">is</span> <span m="1985860">only</span> <span m="1986230">going</span> <span m="1986660">to</span> <span m="1987420">be</span> <span m="1987790">participating</span> <span m="1988670">in</span> <span m="1988750">the</span> <span m="1988870">wires</span> <span m="1989400">off</span> <span m="1989780">the</span> <span m="1990840">i-th</span> <span m="1991203">throw</span> <span m="1992320">and</span> <span m="1992710">the</span> <span m="1992990">P-i-th</span> <span m="1994080">column,</span> <span m="1995510">then</span> <span m="1995700">I</span> <span m="1995760">know</span> <span m="1996210">that</span> <span m="1997580">every</span> <span m="1998180">wire</span> <span m="1998870">will</span> <span m="1999170">only</span> <span m="1999540">get</span> <span m="2000420">traveled</span> <span m="2000850">over</span> <span m="2001680">twice</span> <span m="2004120">by</span> <span m="2004620">a packet.</span> <span m="2004780">This</span> <span m="2004940">could</span> <span m="2005140">either</span> <span m="2005440">be</span> <span m="2006540">a</span> <span m="2006640">packet</span> <span m="2007060">that</span> <span m="2007240">goes</span> <span m="2007560">into</span> <span m="2007810">this</span> <span m="2008020">direction</span> <span m="2009160">or--</span> <span m="2010110">so</span> <span m="2010260">a</span> <span m="2010540">switch</span> <span m="2011010">will</span> <span m="2011100">be</span> <span m="2011200">accessed</span> <span m="2011810">at</span> <span m="2012000">most</span> <span m="2012360">twice.</span> <span m="2012650">A</span> <span m="2012940">switch</span> <span m="2013340">can</span> <span m="2013510">either</span> <span m="2013770">receive</span> <span m="2014160">a</span> <span m="2014240">packet</span> <span m="2014990">from</span> <span m="2015220">this</span> <span m="2015410">direction</span> <span m="2016160">or</span> <span m="2016370">receive</span> <span m="2016630">a</span> <span m="2016730">packet</span> <span m="2017170">from</span> <span m="2020010">the</span> <span m="2020130">upper</span> <span m="2020350">part.</span> <span m="2022140">So</span> <span m="2022820">that will</span> <span m="2022990">be a</span> <span m="2023160">really</span> <span m="2023440">good</span> <span m="2023680">idea.</span> <span m="2024090">So</span> <span m="2024240">let's</span> <span m="2024860">define</span> <span m="2025350">that.</span> </p>
<p><span m="2026220">So</span> <span m="2028790">we</span> <span m="2028930">say</span> <span m="2029210">that</span> <span m="2030080">in</span> <span m="2030250">our</span> <span m="2030410">solution,</span> <span m="2031420">we</span> <span m="2031560">will</span> <span m="2031660">design</span> <span m="2032090">it such</span> <span m="2032460">that</span> <span m="2032650">the</span> <span m="2032750">path</span> <span m="2033300">from</span> <span m="2034500">input</span> <span m="2035020">i</span> <span m="2036150">is</span> <span m="2036530">actually</span> <span m="2037020">going</span> <span m="2037310">to</span> <span m="2037430">be</span> <span m="2037650">rightward</span> <span m="2042300">to</span> <span m="2043050">column</span> <span m="2046410">pi</span> <span m="2047030">i</span> <span m="2048139">and</span> <span m="2048400">then</span> <span m="2048590">downward</span> <span m="2049630">to</span> <span m="2049909">the</span> <span m="2050040">output--</span> <span m="2052830">so</span> <span m="2053090">downward</span> <span m="2054929">to</span> <span m="2056270">output</span> <span m="2059250">by</span> <span m="2060090">i.</span> <span m="2061280">So</span> <span m="2061460">this</span> <span m="2061719">is</span> <span m="2063810">a</span> <span m="2063940">really</span> <span m="2065290">good</span> <span m="2066580">solution</span> <span m="2067340">to</span> <span m="2067460">the</span> <span m="2067570">routing</span> <span m="2067870">problem</span> <span m="2068870">because</span> <span m="2069290">now,</span> <span m="2070650">we</span> <span m="2070820">can</span> <span m="2071469">continue</span> <span m="2071840">our</span> <span m="2071969">proof</span> <span m="2072250">as</span> <span m="2072409">follows.</span> </p>
<p><span m="2072820">We</span> <span m="2072960">just</span> <span m="2073159">say,</span> <span m="2073760">well,</span> <span m="2077600">if</span> <span m="2077760">you</span> <span m="2077889">look</span> <span m="2078290">at</span> <span m="2078639">the</span> <span m="2078750">switch</span> <span m="2080489">in</span> <span m="2081139">row</span> <span m="2082469">i</span> <span m="2083440">and</span> <span m="2084260">column</span> <span m="2085949">pi</span> <span m="2086260">i,</span> <span m="2089489">well,</span> <span m="2089650">this</span> <span m="2089909">one</span> <span m="2090460">actually</span> <span m="2090969">transmits</span> <span m="2091730">at</span> <span m="2091900">most</span> <span m="2092489">two</span> <span m="2092690">packets</span> <span m="2101060">because</span> <span m="2102370">a</span> <span m="2102520">packet</span> <span m="2102840">can</span> <span m="2103000">only</span> <span m="2103260">come</span> <span m="2103500">from</span> <span m="2106060">the</span> <span m="2107650">left</span> <span m="2108260">or</span> <span m="2108440">it's</span> <span m="2108530">going</span> <span m="2108650">to</span> <span m="2108700">go</span> <span m="2108930">from</span> <span m="2109600">the</span> <span m="2109720">top.</span> <span m="2110880">So</span> <span m="2111920">either</span> <span m="2112620">one</span> <span m="2112880">of</span> <span m="2113010">the</span> <span m="2113130">two--</span> <span m="2114640">at</span> <span m="2114750">most,</span> <span m="2115030">those</span> <span m="2115300">two</span> <span m="2115560">packets</span> <span m="2115810">will</span> <span m="2116060">go</span> <span m="2116370">through</span> <span m="2116810">the</span> <span m="2116900">switch.</span> <span m="2117990">So</span> <span m="2118150">this</span> <span m="2118340">shows</span> <span m="2119070">that</span> <span m="2119240">we</span> <span m="2119400">have</span> <span m="2121330">a</span> <span m="2121560">congestion</span> <span m="2122490">of</span> <span m="2123210">at</span> <span m="2123370">most</span> <span m="2123740">two</span> <span m="2124050">for</span> <span m="2124410">any</span> <span m="2124660">permutation.</span> </p>
<p><span m="2125990">And</span> <span m="2126220">in</span> <span m="2126360">order</span> <span m="2126540">to</span> <span m="2126680">prove</span> <span m="2127020">equality,</span> <span m="2127740">because</span> <span m="2128000">that's</span> <span m="2128190">really</span> <span m="2128360">what</span> <span m="2128480">the</span> <span m="2128580">theorem</span> <span m="2128900">says,</span> <span m="2129640">we</span> <span m="2129760">also</span> <span m="2129950">have</span> <span m="2130060">to</span> <span m="2130160">show</span> <span m="2130370">that</span> <span m="2130450">there</span> <span m="2130620">exists</span> <span m="2131510">a</span> <span m="2131620">permutation</span> <span m="2132860">that</span> <span m="2133480">achieves</span> <span m="2134510">a</span> <span m="2134620">congestion</span> <span m="2135170">of</span> <span m="2135320">two.</span> <span m="2136300">And</span> <span m="2136520">that</span> <span m="2136710">is</span> <span m="2137050">pretty</span> <span m="2137310">straightforward.</span> <span m="2138340">We</span> <span m="2138440">can,</span> <span m="2139100">for</span> <span m="2139230">example</span> <span m="2140590">use</span> <span m="2141200">a</span> <span m="2141290">specific</span> <span m="2142520">permutation</span> <span m="2143310">that</span> <span m="2143730">maps</span> <span m="2144830">zero</span> <span m="2145680">to</span> <span m="2145830">zero</span> <span m="2146880">and</span> <span m="2147720">maps</span> <span m="2148820">n</span> <span m="2149070">minus</span> <span m="2149470">1</span> <span m="2150020">to</span> <span m="2150300">n</span> <span m="2150440">minus</span> <span m="2150820">1.</span> </p>
<p><span m="2151980">Well,</span> <span m="2152120">for</span> <span m="2152280">this</span> <span m="2152470">particular</span> <span m="2152880">permutation,</span> <span m="2153940">when</span> <span m="2154110">we</span> <span m="2154240">look</span> <span m="2154470">at</span> <span m="2155570">the</span> <span m="2156190">picture</span> <span m="2156520">over</span> <span m="2156750">here,</span> <span m="2157500">we</span> <span m="2158170">see</span> <span m="2158240">that</span> <span m="2158570">input</span> <span m="2158880">zero</span> <span m="2159670">needs</span> <span m="2159940">to</span> <span m="2160060">go</span> <span m="2160760">to</span> <span m="2160930">output</span> <span m="2161280">zero.</span> <span m="2163630">We</span> <span m="2163750">also</span> <span m="2163950">see</span> <span m="2164300">that</span> <span m="2164680">this</span> <span m="2165830">lowest</span> <span m="2168150">input,</span> <span m="2168500">input</span> <span m="2168850">three,</span> <span m="2169370">needs to</span> <span m="2169510">travel</span> <span m="2170050">all</span> <span m="2170260">the</span> <span m="2170350">way</span> <span m="2170570">up</span> <span m="2170660">to</span> <span m="2170820">here.</span> <span m="2172370">But</span> <span m="2172590">it's</span> <span m="2172840">clear</span> <span m="2173160">that</span> <span m="2173310">the</span> <span m="2173410">packet</span> <span m="2173700">that</span> <span m="2173830">needs</span> <span m="2174140">to</span> <span m="2174620">go</span> <span m="2174970">over</span> <span m="2175120">here</span> <span m="2175570">needs</span> <span m="2175740">to</span> <span m="2175860">travel</span> <span m="2176180">through</span> <span m="2176370">that</span> <span m="2176560">switch</span> <span m="2177800">in</span> <span m="2178480">the</span> <span m="2178820">lower</span> <span m="2182190">left</span> <span m="2182770">bottom</span> <span m="2183140">corner.</span> <span m="2184360">And</span> <span m="2185330">the</span> <span m="2185430">input</span> <span m="2186260">three</span> <span m="2186640">also</span> <span m="2186910">needs</span> <span m="2187050">to</span> <span m="2187170">travel</span> <span m="2187490">through</span> <span m="2187710">that.</span> <span m="2188000">So</span> <span m="2188230">here,</span> <span m="2188470">we</span> <span m="2188860">clearly</span> <span m="2189170">see</span> <span m="2189420">that we</span> <span m="2189600">you</span> <span m="2189670">have a</span> <span m="2189990">congestion</span> <span m="2191460">of</span> <span m="2191700">two.</span> </p>
<p><span m="2192510">So</span> <span m="2192670">now,</span> <span m="2192840">the</span> <span m="2192960">proof</span> <span m="2193260">is</span> <span m="2193430">complete</span> <span m="2194270">because</span> <span m="2195020">we</span> <span m="2195140">have</span> <span m="2195240">shown</span> <span m="2196560">this</span> <span m="2196870">upper</span> <span m="2197090">bound.</span> <span m="2197460">So</span> <span m="2197690">for</span> <span m="2197940">any</span> <span m="2198210">permutation,</span> <span m="2199290">the</span> <span m="2199390">congestion</span> <span m="2199920">is</span> <span m="2200080">at</span> <span m="2200170">most</span> <span m="2200450">two</span> <span m="2201170">and</span> <span m="2201390">we</span> <span m="2202680">see</span> <span m="2203000">that</span> <span m="2204430">this</span> <span m="2204940">specific</span> <span m="2205620">permutation</span> <span m="2206640">achieves</span> <span m="2208170">this</span> <span m="2208990">congestion.</span> <span m="2209840">So</span> <span m="2209990">this</span> <span m="2210150">is</span> <span m="2210280">the</span> <span m="2210410">end</span> <span m="2210600">of</span> <span m="2210710">this</span> <span m="2211270">proof.</span> <span m="2212840">So</span> <span m="2212970">that's</span> <span m="2213170">great.</span> <span m="2214140">So</span> <span m="2214310">now,</span> <span m="2214640">what</span> <span m="2214770">we'd</span> <span m="2214890">like</span> <span m="2215080">to</span> <span m="2215200">do</span> <span m="2215350">is</span> <span m="2215430">we'd</span> <span m="2215570">like</span> <span m="2215740">to</span> <span m="2215850">combine</span> <span m="2216760">these</span> <span m="2217100">two</span> <span m="2217990">networks</span> <span m="2218660">and</span> <span m="2218760">see</span> <span m="2218920">what</span> <span m="2219000">we</span> <span m="2219150">can</span> <span m="2219350">learn</span> <span m="2219620">from</span> <span m="2219830">both.</span> <span m="2221340">So</span> <span m="2222230">now,</span> <span m="2222390">we'll</span> <span m="2222510">be</span> <span m="2223260">taking</span> <span m="2223710">out</span> <span m="2224380">a lot</span> <span m="2224590">of</span> <span m="2226770">chalk</span> <span m="2227190">over</span> <span m="2227460">here.</span> </p>
<p><span m="2229910">So</span> <span m="2231270">the</span> <span m="2231440">idea</span> <span m="2231810">is</span> <span m="2231970">to</span> <span m="2232110">construct</span> <span m="2232590">a</span> <span m="2232660">butterfly</span> <span m="2233200">network</span> <span m="2234600">and</span> <span m="2237470">I</span> <span m="2237620">will</span> <span m="2238050">draw</span> <span m="2238400">it</span> <span m="2238510">in</span> <span m="2238630">such a</span> <span m="2238950">way</span> <span m="2239550">that</span> <span m="2239700">you</span> <span m="2239790">can</span> <span m="2239930">see</span> <span m="2240110">the</span> <span m="2240220">recursive</span> <span m="2240710">structure.</span> <span m="2245620">The</span> <span m="2245800">idea</span> <span m="2246120">is</span> <span m="2246320">to</span> <span m="2253590">do</span> <span m="2253710">the</span> <span m="2253810">following</span> <span m="2254220">thing.</span> <span m="2254630">So</span> <span m="2255320">let</span> <span m="2255510">me</span> <span m="2255880">see</span> <span m="2256170">how</span> <span m="2256370">I</span> <span m="2256440">can</span> <span m="2256670">do</span> <span m="2256830">this</span> <span m="2257070">the</span> <span m="2257140">best.</span> <span m="2259532">So</span> <span m="2260910">I will</span> <span m="2261090">just</span> <span m="2261260">do</span> <span m="2261360">the</span> <span m="2262420">top</span> <span m="2262660">line</span> <span m="2262890">first</span> <span m="2263280">and I have</span> <span m="2263580">the</span> <span m="2263690">spacing.</span> <span m="2265240">So</span> <span m="2265440">we</span> <span m="2265560">have</span> <span m="2266000">input</span> <span m="2266360">zero,</span> <span m="2267500">a</span> <span m="2267640">terminal,</span> <span m="2268980">we</span> <span m="2269110">have</span> <span m="2269330">a</span> <span m="2269430">switch,</span> <span m="2270710">we</span> <span m="2270800">have</span> <span m="2271020">a</span> <span m="2271130">switch,</span> <span m="2273201">we</span> <span m="2273610">have</span> <span m="2273870">a switch,</span> <span m="2275270">and</span> <span m="2275650">another</span> <span m="2275840">one,</span> <span m="2278230">and</span> <span m="2278580">here,</span> <span m="2278860">we</span> <span m="2279030">have</span> <span m="2279650">the</span> <span m="2279870">output</span> <span m="2281400">zero.</span> </p>
<p><span m="2282900">So</span> <span m="2283140">the</span> <span m="2283230">whole</span> <span m="2283450">idea</span> <span m="2283890">is</span> <span m="2286240">that</span> <span m="2286440">I'm</span> <span m="2286520">going</span> <span m="2286860">to</span> <span m="2287010">combine</span> <span m="2288330">every</span> <span m="2289170">two</span> <span m="2289490">outputs</span> <span m="2291490">by</span> <span m="2291640">using</span> <span m="2292760">a</span> <span m="2292860">small</span> <span m="2293250">butterfly</span> <span m="2293800">structure.</span> <span m="2295820">So</span> <span m="2296060">we</span> <span m="2296150">have</span> <span m="2296750">two,</span> <span m="2298530">output</span> <span m="2298920">three,</span> <span m="2302920">output</span> <span m="2303490">four--</span> <span m="2306780">actually,</span> <span m="2307160">I</span> <span m="2307190">need</span> <span m="2307370">a</span> <span m="2307390">little</span> <span m="2307580">bit</span> <span m="2307720">more</span> <span m="2307880">space.</span> <span m="2313001">Do it</span> <span m="2313490">once</span> <span m="2313730">more,</span> <span m="2317820">output</span> <span m="2318030">one,</span> <span m="2320070">two,</span> <span m="2321730">three,</span> <span m="2323430">four,</span> <span m="2325060">five,</span> <span m="2326460">six,</span> <span m="2327080">and</span> <span m="2327260">a</span> <span m="2327350">last</span> <span m="2327690">one,</span> <span m="2327860">seven.</span> <span m="2329340">This</span> <span m="2329500">is</span> <span m="2329660">going</span> <span m="2329860">to</span> <span m="2329950">be</span> <span m="2330090">pretty</span> <span m="2330430">tight</span> <span m="2330730">on</span> <span m="2330810">the</span> <span m="2330880">board.</span> </p>
<p><span m="2331900">So</span> <span m="2332990">what's</span> <span m="2333210">happening</span> <span m="2333680">is</span> <span m="2333830">this.</span> <span m="2334290">So</span> <span m="2335440">these</span> <span m="2335670">are</span> <span m="2335740">all</span> <span m="2335950">connected,</span> <span m="2336460">of</span> <span m="2336620">course,</span> <span m="2336940">to</span> <span m="2337070">switches.</span> <span m="2337670">The</span> <span m="2337770">switches</span> <span m="2338300">output</span> <span m="2338790">those.</span> <span m="2345540">And</span> <span m="2347000">the</span> <span m="2347160">idea</span> <span m="2347580">is</span> <span m="2347940">that</span> <span m="2348870">we</span> <span m="2349100">create</span> <span m="2349620">the</span> <span m="2349720">following</span> <span m="2350130">structure.</span> <span m="2351490">This</span> <span m="2351720">switch</span> <span m="2352050">can</span> <span m="2352250">either</span> <span m="2352530">forward</span> <span m="2353180">it</span> <span m="2353440">over</span> <span m="2353750">here</span> <span m="2354340">or</span> <span m="2354570">it can</span> <span m="2354810">cross</span> <span m="2355180">it</span> <span m="2355310">over</span> <span m="2355820">to</span> <span m="2355970">this</span> <span m="2356220">particular</span> <span m="2356710">line.</span> <span m="2357430">And</span> <span m="2357570">this</span> <span m="2357790">switch</span> <span m="2358000">can</span> <span m="2358220">either</span> <span m="2359180">forward</span> <span m="2359560">it</span> <span m="2360000">or</span> <span m="2360140">cross</span> <span m="2360530">it over</span> <span m="2360720">to</span> <span m="2360890">this</span> <span m="2361170">line.</span> <span m="2361410">So</span> <span m="2361420">this</span> <span m="2361600">is</span> <span m="2361710">a</span> <span m="2361770">very</span> <span m="2362120">small</span> <span m="2362530">butterfly</span> <span m="2363150">structure.</span> <span m="2364180">Here, we have</span> <span m="2364520">two</span> <span m="2364750">inputs</span> <span m="2365670">and two</span> <span m="2366040">outputs.</span> <span m="2367980">And</span> <span m="2368380">we will</span> <span m="2368710">repeat</span> <span m="2369090">this</span> <span m="2369270">process</span> <span m="2370220">and</span> <span m="2370370">we'll</span> <span m="2370480">do</span> <span m="2370630">the</span> <span m="2370750">same</span> <span m="2371340">on</span> <span m="2371540">each</span> <span m="2371760">of</span> <span m="2371890">these</span> <span m="2372090">other</span> <span m="2372260">levels.</span> <span m="2373420">So</span> <span m="2373880">we</span> <span m="2374320">forward</span> <span m="2374950">those</span> <span m="2375710">or</span> <span m="2375960">we</span> <span m="2376080">cross</span> <span m="2376450">them,</span> <span m="2383500">like</span> <span m="2383700">this.</span> </p>
<p><span m="2388460">And</span> <span m="2390020">now</span> <span m="2390250">that</span> <span m="2390300">we have</span> <span m="2390700">constructed</span> <span m="2392050">all</span> <span m="2392260">these</span> <span m="2392400">smaller</span> <span m="2392710">butterfly</span> <span m="2393190">structures,</span> <span m="2394260">we</span> <span m="2394390">can</span> <span m="2394580">start</span> <span m="2394930">to</span> <span m="2395030">combine</span> <span m="2395600">two</span> <span m="2395840">butterfly</span> <span m="2396360">structures</span> <span m="2397000">together</span> <span m="2397470">in the</span> <span m="2397660">bigger</span> <span m="2397970">one.</span> <span m="2399020">So</span> <span m="2399270">here,</span> <span m="2399450">we</span> <span m="2399550">had</span> <span m="2399770">two</span> <span m="2400090">outputs</span> <span m="2400520">that</span> <span m="2400610">we</span> <span m="2400720">combined</span> <span m="2401180">in</span> <span m="2401290">a</span> <span m="2402100">butterfly</span> <span m="2402530">structure.</span> <span m="2403030">Now,</span> <span m="2403170">we</span> <span m="2403270">use</span> <span m="2403500">two</span> <span m="2403680">butterfly</span> <span m="2404210">structures</span> <span m="2404700">that</span> <span m="2405490">we</span> <span m="2408120">put</span> <span m="2408390">into</span> <span m="2408670">a</span> <span m="2408720">bigger</span> <span m="2409510">version.</span> <span m="2409930">So</span> <span m="2410080">how</span> <span m="2410200">do</span> <span m="2410300">we</span> <span m="2410410">do</span> <span m="2410600">this?</span> <span m="2411580">Well,</span> <span m="2412720">we</span> <span m="2412920">have</span> <span m="2414230">that</span> <span m="2414440">the</span> <span m="2414610">upper</span> <span m="2414940">half</span> <span m="2415430">over</span> <span m="2415760">here</span> <span m="2416130">can</span> <span m="2416310">either</span> <span m="2416570">forward</span> <span m="2417390">those</span> <span m="2419010">packets</span> <span m="2419830">or</span> <span m="2420500">cross</span> <span m="2420820">them</span> <span m="2420960">over</span> <span m="2421300">to</span> <span m="2421470">the</span> <span m="2421640">bottom</span> <span m="2422090">part</span> <span m="2422990">butterfly</span> <span m="2423520">structure.</span> </p>
<p><span m="2425510">So</span> <span m="2426300">for</span> <span m="2426480">these,</span> <span m="2427020">we</span> <span m="2427110">can</span> <span m="2427300">either</span> <span m="2427560">forward</span> <span m="2428120">them</span> <span m="2428620">straight</span> <span m="2429160">on</span> <span m="2430880">or</span> <span m="2431900">we</span> <span m="2432020">can</span> <span m="2432180">go</span> <span m="2432400">to</span> <span m="2432580">the</span> <span m="2432850">top</span> <span m="2434900">butterfly.</span> <span m="2436290">So</span> <span m="2436500">you</span> <span m="2436620">see</span> <span m="2437370">that</span> <span m="2437980">these</span> <span m="2438240">two</span> <span m="2438390">inputs,</span> <span m="2439160">these</span> <span m="2439370">two</span> <span m="2439520">switches,</span> <span m="2440650">either</span> <span m="2441700">can</span> <span m="2442000">forward</span> <span m="2442380">packets</span> <span m="2442940">to</span> <span m="2443380">this</span> <span m="2443840">sub-butterfly</span> <span m="2445180">network</span> <span m="2446330">or</span> <span m="2447140">to</span> <span m="2447430">the</span> <span m="2447730">top</span> <span m="2448180">butterfly</span> <span m="2448710">network.</span> <span m="2450220">Now,</span> <span m="2450280">we'll</span> <span m="2450430">continue</span> <span m="2450830">this</span> <span m="2451030">process</span> <span m="2453700">and</span> <span m="2454270">for</span> <span m="2454440">these,</span> <span m="2454660">you'll</span> <span m="2454840">do</span> <span m="2454960">the</span> <span m="2455080">same.</span> <span m="2455800">So</span> <span m="2455990">we</span> <span m="2456080">can</span> <span m="2456290">either</span> <span m="2457020">go</span> <span m="2457200">straight</span> <span m="2458640">or</span> <span m="2458840">we</span> <span m="2458960">go</span> <span m="2459160">down.</span> <span m="2461490">And</span> <span m="2461670">over</span> <span m="2461990">here,</span> <span m="2462320">we</span> <span m="2462410">can</span> <span m="2462550">go</span> <span m="2462740">straight</span> <span m="2463520">or</span> <span m="2463750">we</span> <span m="2463850">can</span> <span m="2464040">go</span> <span m="2464400">to</span> <span m="2464510">the</span> <span m="2464670">top</span> <span m="2466150">butterfly</span> <span m="2466370">network.</span> </p>
<p><span m="2468320">Well,</span> <span m="2468500">now</span> <span m="2468730">we</span> <span m="2469650">have</span> <span m="2470690">the</span> <span m="2470810">final</span> <span m="2472530">part</span> <span m="2472940">where we</span> <span m="2473220">combine</span> <span m="2474310">essentially</span> <span m="2475520">these</span> <span m="2476600">two</span> <span m="2476960">butterfly</span> <span m="2477430">networks.</span> <span m="2479170">We</span> <span m="2479280">have</span> <span m="2479570">two</span> <span m="2479840">butterfly</span> <span m="2480560">networks</span> <span m="2480810">created</span> <span m="2481280">here</span> <span m="2481620">now</span> <span m="2482460">composed</span> <span m="2482910">again of</span> <span m="2483310">smaller</span> <span m="2483610">ones</span> <span m="2483940">and</span> <span m="2484170">these</span> <span m="2484390">two</span> <span m="2484560">are</span> <span m="2484710">being</span> <span m="2484910">composed</span> <span m="2485470">to</span> <span m="2485760">this</span> <span m="2485970">bigger</span> <span m="2486230">butterfly</span> <span m="2486700">network.</span> <span m="2487660">Again,</span> <span m="2488590">we</span> <span m="2488730">take</span> <span m="2489900">these</span> <span m="2490260">four</span> <span m="2491880">switches.</span> <span m="2492410">They</span> <span m="2492540">can</span> <span m="2492780">route</span> <span m="2493420">their</span> <span m="2493580">packets</span> <span m="2494020">forward</span> <span m="2494620">to</span> <span m="2495530">the</span> <span m="2495710">top</span> <span m="2496660">butterfly</span> <span m="2497380">sub-network</span> <span m="2498800">or</span> <span m="2499020">to</span> <span m="2499180">the</span> <span m="2499300">bottom</span> <span m="2499665">one.</span> </p>
<p><span m="2500250">So</span> <span m="2501140">they</span> <span m="2501240">can</span> <span m="2501440">either</span> <span m="2501640">go</span> <span m="2501850">straight</span> <span m="2502280">ahead</span> <span m="2503650">or</span> <span m="2503880">this</span> <span m="2504110">one</span> <span m="2504780">can</span> <span m="2505030">connect</span> <span m="2505520">to</span> <span m="2506380">the</span> <span m="2506480">first</span> <span m="2506870">over</span> <span m="2507160">here,</span> <span m="2508480">this</span> <span m="2508680">one</span> <span m="2508770">to</span> <span m="2508870">the</span> <span m="2508970">second,</span> <span m="2509950">to</span> <span m="2510050">the</span> <span m="2510160">third,</span> <span m="2511280">and</span> <span m="2511470">this</span> <span m="2511600">to</span> <span m="2511720">the</span> <span m="2511840">fourth.</span> <span m="2513410">And in</span> <span m="2513560">the</span> <span m="2513690">same</span> <span m="2513940">style,</span> <span m="2514360">these</span> <span m="2515230">can</span> <span m="2515380">forward</span> <span m="2515870">them</span> <span m="2516440">straight</span> <span m="2518410">like</span> <span m="2518650">this</span> <span m="2519500">and</span> <span m="2519630">then</span> <span m="2519820">go</span> <span m="2520130">up</span> <span m="2527000">like</span> <span m="2527220">this.</span> <span m="2529760">And</span> <span m="2530020">these</span> <span m="2530220">are</span> <span m="2530290">all</span> <span m="2530430">connected</span> <span m="2531250">because</span> <span m="2531540">in</span> <span m="2531600">this</span> <span m="2531740">example,</span> <span m="2532210">let's</span> <span m="2532430">just</span> <span m="2532650">have</span> <span m="2532790">an</span> <span m="2532940">eight</span> <span m="2533150">by</span> <span m="2533270">eight</span> <span m="2534760">network,</span> <span m="2535300">butterfly</span> <span m="2535700">network.</span> <span m="2536720">We</span> <span m="2536850">have</span> <span m="2537430">input</span> <span m="2540720">zero</span> <span m="2541150">to</span> <span m="2541340">seven.</span> </p>
<p><span m="2552820">So</span> <span m="2552990">this</span> <span m="2553220">is</span> <span m="2553320">the</span> <span m="2553640">butterfly</span> <span m="2554090">network.</span> <span m="2554440">In a way,</span> <span m="2554690">what</span> <span m="2554870">you</span> <span m="2554970">can</span> <span m="2555130">see</span> <span m="2555410">here</span> <span m="2555670">is you</span> <span m="2555770">can</span> <span m="2555900">see</span> <span m="2556650">sort</span> <span m="2556820">of</span> <span m="2556910">the</span> <span m="2557030">two-dimensional</span> <span m="2557760">structure,</span> <span m="2558950">like</span> <span m="2559210">we</span> <span m="2559320">have</span> <span m="2559980">rows</span> <span m="2560440">and</span> <span m="2560700">columns.</span> <span m="2561640">At</span> <span m="2561750">the</span> <span m="2561830">same</span> <span m="2562140">time,</span> <span m="2562460">we</span> <span m="2562580">can</span> <span m="2562770">also</span> <span m="2562980">see</span> <span m="2563320">this</span> <span m="2563790">binary</span> <span m="2564660">sort</span> <span m="2564960">of</span> <span m="2565140">tree</span> <span m="2566330">feeling</span> <span m="2566810">we</span> <span m="2566950">get</span> <span m="2567320">from</span> <span m="2567600">it,</span> <span m="2567860">which</span> <span m="2568030">is</span> <span m="2568230">that</span> <span m="2568810">a</span> <span m="2569000">switch</span> <span m="2569530">can</span> <span m="2570160">forward</span> <span m="2571860">sort</span> <span m="2572080">of</span> <span m="2573040">its</span> <span m="2573340">packets</span> <span m="2573910">to</span> <span m="2574250">either,</span> <span m="2574920">say,</span> <span m="2575650">the</span> <span m="2575820">top</span> <span m="2577340">butterfly</span> <span m="2578040">or</span> <span m="2578270">the</span> <span m="2578530">bottom</span> <span m="2578890">butterfly.</span> <span m="2579420">So</span> <span m="2579590">there's</span> <span m="2579790">a</span> <span m="2579850">split</span> <span m="2580260">in</span> <span m="2580390">two.</span> <span m="2581900">The</span> <span m="2582010">same</span> <span m="2582240">for</span> <span m="2582400">this</span> <span m="2582620">one,</span> <span m="2582800">right?</span> <span m="2583270">This</span> <span m="2583480">one</span> <span m="2584180">goes</span> <span m="2584390">either</span> <span m="2584710">to</span> <span m="2585010">this</span> <span m="2585270">butterfly</span> <span m="2585520">network</span> <span m="2587020">or</span> <span m="2587200">it</span> <span m="2587310">goes</span> <span m="2587620">to</span> <span m="2588000">this</span> <span m="2588540">butterfly</span> <span m="2588790">network.</span> <span m="2589850">So</span> <span m="2590150">you</span> <span m="2590450">have</span> <span m="2590640">this</span> <span m="2590830">tree</span> <span m="2591070">structure</span> <span m="2591560">sort</span> <span m="2591810">of</span> <span m="2592200">embedded</span> <span m="2592870">in</span> <span m="2593100">this</span> <span m="2593320">two-dimensional</span> <span m="2594050">structure.</span> </p>
<p><span m="2596000">So</span> <span m="2596470">what</span> <span m="2596590">are</span> <span m="2596650">the</span> <span m="2596750">properties</span> <span m="2597300">of</span> <span m="2597420">this</span> <span m="2597650">one?</span> <span m="2598580">So</span> <span m="2598620">let</span> <span m="2598710">me</span> <span m="2598820">first</span> <span m="2599410">define</span> <span m="2605370">in</span> <span m="2605440">more</span> <span m="2605620">formal</span> <span m="2606570">mathematics</span> <span m="2608690">how</span> <span m="2608880">the</span> <span m="2608990">switches</span> <span m="2610130">route their</span> <span m="2610510">packets,</span> <span m="2611660">so</span> <span m="2611900">how</span> <span m="2612090">the</span> <span m="2612220">connections</span> <span m="2612810">are.</span> <span m="2615630">So</span> <span m="2615850">in order</span> <span m="2616100">to</span> <span m="2616190">do</span> <span m="2616350">that,</span> <span m="2616780">we are going</span> <span m="2617090">to</span> <span m="2618130">label</span> <span m="2618650">each</span> <span m="2619140">switch.</span> <span m="2622720">And</span> <span m="2622870">the</span> <span m="2623010">idea</span> <span m="2623295">is</span> <span m="2623580">that</span> <span m="2623940">we're</span> <span m="2624010">going</span> <span m="2624260">to</span> <span m="2624380">label</span> <span m="2624720">it</span> <span m="2625070">by</span> <span m="2625720">its</span> <span m="2626310">row</span> <span m="2626880">and</span> <span m="2627030">by its</span> <span m="2627330">column.</span> </p>
<p><span m="2628480">So</span> <span m="2629470">we</span> <span m="2629600">will</span> <span m="2629790">have--</span> <span m="2633030">the</span> <span m="2633140">columns</span> <span m="2633510">are</span> <span m="2633580">numbered</span> <span m="2634010">by</span> <span m="2634570">level</span> <span m="2634940">zero,</span> <span m="2636270">level</span> <span m="2636720">one,</span> <span m="2637580">level</span> <span m="2638070">two,</span> <span m="2638960">level</span> <span m="2639300">three,</span> <span m="2641530">yes?</span> <span m="2642580">And</span> <span m="2643740">the</span> <span m="2643970">rows</span> <span m="2644770">are</span> <span m="2645460">these</span> <span m="2645780">integers,</span> <span m="2646120">but</span> <span m="2646570">we are</span> <span m="2646680">going</span> <span m="2646990">to</span> <span m="2647590">represent</span> <span m="2648200">them</span> <span m="2648420">by</span> <span m="2648540">binary</span> <span m="2649020">numbers.</span> <span m="2649930">So</span> <span m="2650200">zero</span> <span m="2650660">would</span> <span m="2650880">be</span> <span m="2651970">000,</span> <span m="2653437">001,</span> <span m="2654415">010,</span> <span m="2655882">011--</span> <span m="2658318">oops--</span> <span m="2658804">100,</span> <span m="2660748">101,</span> <span m="2662210">and</span> <span m="2662420">then</span> <span m="2662550">we</span> <span m="2662700">got</span> <span m="2665210">110</span> <span m="2666830">and</span> <span m="2667030">111.</span> </p>
<p><span m="2669320">So</span> <span m="2670070">for</span> <span m="2670230">example,</span> <span m="2670710">this</span> <span m="2670830">particular</span> <span m="2671230">switch</span> <span m="2672030">would</span> <span m="2672230">be</span> <span m="2672370">labeled</span> <span m="2672850">by</span> <span m="2673080">these</span> <span m="2673590">three</span> <span m="2674140">bits,</span> <span m="2674660">001,</span> <span m="2675800">and the</span> <span m="2676100">integer</span> <span m="2676550">number,</span> <span m="2676900">1.</span> <span m="2678040">This</span> <span m="2678270">one</span> <span m="2678570">would</span> <span m="2678770">be</span> <span m="2679810">011</span> <span m="2682060">and</span> <span m="2683120">its</span> <span m="2683490">column</span> <span m="2683860">is</span> <span m="2684850">indexed</span> <span m="2685460">by</span> <span m="2686040">integer</span> <span m="2686470">2.</span> <span m="2688520">So</span> <span m="2689470">a</span> <span m="2689560">switch</span> <span m="2692630">is</span> <span m="2693920">uniquely</span> <span m="2696800">identified</span> <span m="2704010">by</span> <span m="2704990">its</span> <span m="2706190">row</span> <span m="2708440">and</span> <span m="2708670">column.</span> <span m="2712340">We</span> <span m="2712520">will</span> <span m="2712710">have</span> <span m="2713810">b1</span> <span m="2715740">up</span> <span m="2715970">to</span> <span m="2716670">b</span> <span m="2717590">logarithm</span> <span m="2718820">of</span> <span m="2719060">n,</span> <span m="2720350">which</span> <span m="2720690">are</span> <span m="2720860">the</span> <span m="2720930">number</span> <span m="2721250">of</span> <span m="2721540">bits</span> <span m="2722060">to</span> <span m="2722210">represent</span> <span m="2722760">the</span> <span m="2722860">row</span> <span m="2723280">in</span> <span m="2723610">digits,</span> <span m="2724830">and to</span> <span m="2725180">finally</span> <span m="2725670">have</span> <span m="2726340">an</span> <span m="2726510">integer</span> <span m="2726960">l</span> <span m="2727720">and</span> <span m="2727930">this</span> <span m="2728130">we</span> <span m="2728200">will</span> <span m="2728370">call</span> <span m="2728580">the</span> <span m="2728650">level.</span> </p>
<p><span m="2731300">So</span> <span m="2733190">this</span> <span m="2733630">particular</span> <span m="2734030">switch</span> <span m="2734660">either</span> <span m="2734870">directs</span> <span m="2735900">or</span> <span m="2736290">routes</span> <span m="2736740">a</span> <span m="2736910">packet</span> <span m="2737480">to</span> <span m="2737680">the</span> <span m="2737800">switch</span> <span m="2738230">that</span> <span m="2738320">is</span> <span m="2738540">indexed</span> <span m="2739160">by</span> <span m="2741240">b</span> <span m="2741550">one</span> <span m="2742610">up</span> <span m="2742860">to--</span> <span m="2743790">and</span> <span m="2743930">then</span> <span m="2744030">we</span> <span m="2744180">get</span> <span m="2744400">b,</span> <span m="2744950">l</span> <span m="2745270">plus</span> <span m="2745960">1</span> <span m="2746920">and</span> <span m="2747090">we</span> <span m="2747240">take</span> <span m="2747550">its</span> <span m="2747750">complement.</span> <span m="2748590">So</span> <span m="2748820">instead</span> <span m="2749180">of</span> <span m="2749830">if</span> <span m="2750060">b,</span> <span m="2750250">l plus</span> <span m="2750710">n would</span> <span m="2750810">be</span> <span m="2750980">1,</span> <span m="2751570">we</span> <span m="2751720">would</span> <span m="2751890">have</span> <span m="2752040">a</span> <span m="2752070">0</span> <span m="2752470">here.</span> <span m="2753280">If</span> <span m="2753450">it would</span> <span m="2753610">be</span> <span m="2754080">a</span> <span m="2754240">0,</span> <span m="2754540">we</span> <span m="2754670">will</span> <span m="2754790">have</span> <span m="2754980">a 1</span> <span m="2755220">over</span> <span m="2755470">here.</span> <span m="2758370">But</span> <span m="2758590">we</span> <span m="2758800">repeat</span> <span m="2759230">all</span> <span m="2759450">the</span> <span m="2759560">other</span> <span m="2759790">bits</span> <span m="2761150">and</span> <span m="2761340">we</span> <span m="2761450">get</span> <span m="2762270">to</span> <span m="2762340">b</span> <span m="2763180">log</span> <span m="2763520">n.</span> <span m="2765030">And</span> <span m="2765980">it</span> <span m="2766160">routes us</span> <span m="2766610">back</span> <span m="2766870">to</span> <span m="2766970">the</span> <span m="2767060">next</span> <span m="2767360">level.</span> <span m="2767840">So</span> <span m="2767970">we</span> <span m="2768030">will</span> <span m="2768150">have</span> <span m="2768600">l</span> <span m="2768850">plus</span> <span m="2769240">1.</span> </p>
<p><span m="2771480">Another</span> <span m="2771780">possibility</span> <span m="2772650">because</span> <span m="2772930">there are</span> <span m="2773060">two</span> <span m="2773330">outgoing</span> <span m="2773830">edges</span> <span m="2775480">is</span> <span m="2775900">if</span> <span m="2776190">we</span> <span m="2777220">have</span> <span m="2778200">just</span> <span m="2778580">b1</span> <span m="2780250">and</span> <span m="2781100">we</span> <span m="2781270">just</span> <span m="2781600">copy</span> <span m="2782100">b, l</span> <span m="2782540">plus</span> <span m="2782810">1,</span> <span m="2783570">essentially.</span> <span m="2784830">We</span> <span m="2784980">route</span> <span m="2785230">a</span> <span m="2785320">packet</span> <span m="2785650">straightforward.</span> <span m="2786890">We</span> <span m="2786900">don't</span> <span m="2787120">do</span> <span m="2787230">anything</span> <span m="2787630">special.</span> <span m="2789330">We</span> <span m="2789440">get</span> <span m="2789750">b</span> <span m="2790260">log</span> <span m="2790550">n</span> <span m="2790770">over</span> <span m="2791070">here</span> <span m="2792150">and</span> <span m="2792350">then</span> <span m="2792510">to</span> <span m="2792640">the</span> <span m="2792720">next</span> <span m="2793030">level.</span> <span m="2793300">So</span> <span m="2793490">for</span> <span m="2793660">example,</span> <span m="2795560">let's</span> <span m="2795770">see</span> <span m="2795860">where</span> <span m="2795980">we</span> <span m="2796100">can</span> <span m="2796320">see</span> <span m="2796690">how</span> <span m="2796970">this</span> <span m="2797160">works.</span> </p>
<p><span m="2797980">So</span> <span m="2798240">for</span> <span m="2798390">example,</span> <span m="2798980">take</span> <span m="2800110">this</span> <span m="2800350">particular</span> <span m="2800770">switch.</span> <span m="2802210">We</span> <span m="2802430">have</span> <span m="2803980">010.</span> <span m="2805980">So</span> <span m="2806180">it</span> <span m="2806280">can</span> <span m="2806510">either</span> <span m="2806780">go</span> <span m="2806990">straight</span> <span m="2807490">on</span> <span m="2807630">to</span> <span m="2807740">the</span> <span m="2807830">next</span> <span m="2808150">level.</span> <span m="2808670">It</span> <span m="2808850">would</span> <span m="2808990">go</span> <span m="2809190">to</span> <span m="2809320">010</span> <span m="2810470">but</span> <span m="2810640">then</span> <span m="2811400">instead</span> <span m="2811730">of</span> <span m="2811810">level</span> <span m="2812130">one,</span> <span m="2812300">we have</span> <span m="2812510">level</span> <span m="2812840">two,</span> <span m="2813650">which</span> <span m="2813990">is</span> <span m="2814160">the</span> <span m="2814290">right</span> <span m="2815000">edge</span> <span m="2815310">over</span> <span m="2815530">there.</span> <span m="2818070">The</span> <span m="2818210">other</span> <span m="2818380">one</span> <span m="2818970">is</span> <span m="2819210">if</span> <span m="2819360">this</span> <span m="2819540">one</span> <span m="2819680">goes</span> <span m="2820150">up,</span> <span m="2820980">well,</span> <span m="2822810">we</span> <span m="2823010">will</span> <span m="2823180">need</span> <span m="2823390">to</span> <span m="2824140">switch</span> <span m="2825440">the</span> <span m="2828400">first</span> <span m="2831900">bit</span> <span m="2832170">over</span> <span m="2832470">here,</span> <span m="2832970">a</span> <span m="2833180">1.</span> <span m="2834120">We</span> <span m="2834320">swap</span> <span m="2834660">it</span> <span m="2834780">into</span> <span m="2835000">0</span> <span m="2836080">and</span> <span m="2836210">then</span> <span m="2836280">we</span> <span m="2836480">go</span> <span m="2836660">to</span> <span m="2836940">the</span> <span m="2837110">three</span> <span m="2837370">zeros</span> <span m="2837760">over</span> <span m="2838010">here</span> <span m="2839010">and</span> <span m="2839550">we</span> <span m="2839680">go</span> <span m="2839840">to</span> <span m="2839950">the</span> <span m="2840050">next</span> <span m="2840340">level</span> <span m="2840830">and</span> <span m="2840950">that</span> <span m="2841120">would</span> <span m="2841290">be</span> <span m="2841780">this</span> <span m="2841990">particular</span> <span m="2842770">rule.</span> </p>
<p><span m="2844350">So</span> <span m="2844540">what</span> <span m="2844670">we</span> <span m="2844770">can</span> <span m="2844970">do</span> <span m="2845190">here</span> <span m="2845610">is</span> <span m="2845930">to--</span> <span m="2846850">so</span> <span m="2848360">when</span> <span m="2848490">we</span> <span m="2848590">see</span> <span m="2848870">this,</span> <span m="2849650">we</span> <span m="2849770">can</span> <span m="2849930">start</span> <span m="2850310">to</span> <span m="2851310">figure</span> <span m="2851630">out</span> <span m="2851810">how</span> <span m="2852370">we</span> <span m="2852510">can</span> <span m="2852730">direct</span> <span m="2853710">inputs</span> <span m="2854220">to</span> <span m="2854380">outputs.</span> <span m="2856040">So</span> <span m="2856260">let's</span> <span m="2857750">do</span> <span m="2857930">this.</span> <span m="2865410">So</span> <span m="2868950">suppose</span> <span m="2869510">I</span> <span m="2869570">want</span> <span m="2869850">to</span> <span m="2872570">route</span> <span m="2873340">a</span> <span m="2873530">packet</span> <span m="2874120">from</span> <span m="2874430">a</span> <span m="2874480">certain</span> <span m="2874920">input,</span> <span m="2875950">one</span> <span m="2878130">of</span> <span m="2878250">these,</span> <span m="2879610">all</span> <span m="2879860">the</span> <span m="2879970">way</span> <span m="2880450">to</span> <span m="2880860">one</span> <span m="2881400">of</span> <span m="2881560">the</span> <span m="2881710">outputs</span> <span m="2882820">over</span> <span m="2883080">here.</span> <span m="2884430">So</span> <span m="2887320">the</span> <span m="2887420">way</span> <span m="2887570">to</span> <span m="2887740">do</span> <span m="2887920">this</span> <span m="2888470">is</span> <span m="2888730">as</span> <span m="2888860">follows.</span> </p>
<p><span m="2889900">We</span> <span m="2889910">can</span> <span m="2890070">just</span> <span m="2890300">start--</span> <span m="2890950">for</span> <span m="2891130">example,</span> <span m="2891450">I</span> <span m="2891500">want</span> <span m="2891750">to</span> <span m="2892420">go</span> <span m="2892640">from</span> <span m="2892870">switch</span> <span m="2893990">x1</span> <span m="2895610">up</span> <span m="2895850">to x</span> <span m="2896310">log</span> <span m="2896690">n</span> <span m="2899740">comma</span> <span m="2900000">0.</span> <span m="2901120">So</span> <span m="2901310">I</span> <span m="2901410">start</span> <span m="2901970">completely</span> <span m="2903560">at</span> <span m="2903730">the</span> <span m="2903820">left</span> <span m="2904070">over</span> <span m="2904410">here</span> <span m="2904830">and</span> <span m="2904960">I</span> <span m="2904980">want</span> <span m="2905150">to</span> <span m="2905270">go</span> <span m="2905510">somewhere</span> <span m="2906270">of</span> <span m="2906450">my</span> <span m="2906690">choice</span> <span m="2907190">to</span> <span m="2907310">the</span> <span m="2907440">right.</span> <span m="2908780">So</span> <span m="2908970">I want to</span> <span m="2909200">somehow</span> <span m="2910090">move</span> <span m="2911850">all</span> <span m="2912100">the</span> <span m="2912200">way</span> <span m="2912410">to</span> <span m="2913470">some</span> <span m="2913850">other</span> <span m="2914190">row,</span> <span m="2914910">y1</span> <span m="2916660">indexed</span> <span m="2917120">by</span> <span m="2917240">y</span> <span m="2917790">by</span> <span m="2917910">the bit</span> <span m="2918260">pattern,</span> <span m="2918700">y1</span> <span m="2919220">up to</span> <span m="2919320">y</span> <span m="2920030">log</span> <span m="2920360">n,</span> <span m="2921660">but</span> <span m="2921830">now</span> <span m="2922180">at</span> <span m="2922300">the</span> <span m="2922470">very</span> <span m="2922670">last</span> <span m="2923000">level,</span> <span m="2925000">which</span> <span m="2925210">is</span> <span m="2925430">log</span> <span m="2925700">n.</span> </p>
<p><span m="2926670">Well,</span> <span m="2927010">how</span> <span m="2927200">do</span> <span m="2927320">I</span> <span m="2927430">do</span> <span m="2927620">it?</span> <span m="2928980">Well,</span> <span m="2929210">this</span> <span m="2929520">switch,</span> <span m="2929980">I</span> <span m="2930100">can</span> <span m="2930260">use</span> <span m="2930430">that</span> <span m="2930530">rule</span> <span m="2930820">up</span> <span m="2931040">there</span> <span m="2931550">and</span> <span m="2931830">simply</span> <span m="2932640">change</span> <span m="2933150">x1</span> <span m="2933930">to</span> <span m="2934370">y1.</span> <span m="2935380">I</span> <span m="2935480">can</span> <span m="2935670">either</span> <span m="2935910">leave</span> <span m="2936200">x1</span> <span m="2936670">as</span> <span m="2936810">it</span> <span m="2936910">is</span> <span m="2937700">if</span> <span m="2937910">it's</span> <span m="2938110">the</span> <span m="2938170">same</span> <span m="2938490">as</span> <span m="2938610">y1</span> <span m="2939600">or</span> <span m="2939900">I</span> <span m="2939960">can</span> <span m="2940190">swap</span> <span m="2940490">it</span> <span m="2940590">to</span> <span m="2940720">its</span> <span m="2940840">complement</span> <span m="2941840">if</span> <span m="2942050">that's</span> <span m="2942580">the</span> <span m="2942760">value</span> <span m="2943070">of</span> <span m="2943190">y1.</span> <span m="2944570">So</span> <span m="2944790">what I</span> <span m="2944910">can</span> <span m="2945110">do</span> <span m="2945300">is</span> <span m="2945490">I</span> <span m="2945550">can</span> <span m="2945710">just</span> <span m="2945930">simply</span> <span m="2946300">route</span> <span m="2946575">it</span> <span m="2947360">to</span> <span m="2949760">y1.</span> <span m="2950540">And</span> <span m="2950750">then,</span> <span m="2951900">I</span> <span m="2952090">leave all</span> <span m="2952350">the</span> <span m="2952480">other</span> <span m="2952690">bits</span> <span m="2952970">the</span> <span m="2953040">same,</span> <span m="2953950">which</span> <span m="2954300">are</span> <span m="2954470">x2,</span> <span m="2956150">x3,</span> <span m="2956550">all</span> <span m="2956790">the</span> <span m="2956900">way</span> <span m="2957570">up</span> <span m="2957730">to</span> <span m="2957940">x</span> <span m="2958310">log</span> <span m="2958580">n.</span> <span m="2959940">And</span> <span m="2961450">we</span> <span m="2961640">will</span> <span m="2961790">have</span> <span m="2964360">reached</span> <span m="2964810">the</span> <span m="2964900">first</span> <span m="2965260">level.</span> </p>
<p><span m="2967310">Now,</span> <span m="2967480">this</span> <span m="2967850">one</span> <span m="2968430">can</span> <span m="2968620">go</span> <span m="2968880">to--</span> <span m="2969710">well,</span> <span m="2970660">now</span> <span m="2970820">I'm</span> <span m="2970930">going</span> <span m="2971150">to</span> <span m="2971290">swap</span> <span m="2971680">the</span> <span m="2971780">second</span> <span m="2972300">bit</span> <span m="2972560">into</span> <span m="2972840">the</span> <span m="2972970">bit</span> <span m="2973200">of</span> <span m="2973310">my</span> <span m="2973470">choice.</span> <span m="2974420">So</span> <span m="2975220">I</span> <span m="2975340">leave</span> <span m="2975570">all</span> <span m="2975720">the</span> <span m="2975820">other</span> <span m="2976110">bits</span> <span m="2976300">the</span> <span m="2976380">same,</span> <span m="2976790">y1</span> <span m="2977160">the</span> <span m="2977260">same,</span> <span m="2977560">x3,</span> <span m="2978210">all</span> <span m="2978360">the</span> <span m="2978480">others</span> <span m="2978720">the</span> <span m="2978810">same.</span> <span m="2979470">I</span> <span m="2979550">just</span> <span m="2979780">swap</span> <span m="2980550">x2</span> <span m="2981410">into</span> <span m="2981670">y2.</span> <span m="2983900">So</span> <span m="2984310">we</span> <span m="2984430">leave all</span> <span m="2984820">those</span> <span m="2985860">equal</span> <span m="2987810">and</span> <span m="2987960">we</span> <span m="2988040">go</span> <span m="2988220">to</span> <span m="2988330">the</span> <span m="2988430">second</span> <span m="2988770">level.</span> <span m="2990100">And</span> <span m="2990280">then,</span> <span m="2990440">we</span> <span m="2990560">go</span> <span m="2991250">all</span> <span m="2991630">the</span> <span m="2991740">way</span> <span m="2993760">to</span> <span m="2993890">the</span> <span m="2993990">final</span> <span m="2994410">level</span> <span m="2994880">and</span> <span m="2995090">we</span> <span m="2995380">one</span> <span m="2995690">by</span> <span m="2995860">one</span> <span m="2996230">swap</span> <span m="2996550">all</span> <span m="2996780">these</span> <span m="2996970">bits.</span> </p>
<p><span m="2997310">So</span> <span m="2997320">let's</span> <span m="2997480">have</span> <span m="2997610">an</span> <span m="2997680">example.</span> <span m="3000950">Suppose</span> <span m="3001440">I</span> <span m="3001510">want</span> <span m="3001770">to</span> <span m="3001910">connect,</span> <span m="3002950">let's</span> <span m="3003160">say,</span> <span m="3004030">this</span> <span m="3004260">one</span> <span m="3005860">to</span> <span m="3009290">for</span> <span m="3009460">example,</span> <span m="3009945">well,</span> <span m="3010430">let's</span> <span m="3010830">say</span> <span m="3010960">this</span> <span m="3011280">particular</span> <span m="3011750">output.</span> <span m="3013120">So</span> <span m="3015650">what's</span> <span m="3018280">the</span> <span m="3018350">binary</span> <span m="3018780">for</span> <span m="3018940">this</span> <span m="3019170">one?</span> <span m="3019570">This</span> <span m="3019800">is</span> <span m="3020170">actually</span> <span m="3020710">101.</span> <span m="3022900">So</span> <span m="3025260">if</span> <span m="3025530">the</span> <span m="3025630">first</span> <span m="3026050">bit</span> <span m="3026290">is</span> <span m="3026480">different,</span> <span m="3027490">I</span> <span m="3027590">need</span> <span m="3027790">to</span> <span m="3027910">cross.</span> <span m="3029030">And</span> <span m="3029460">otherwise,</span> <span m="3030100">I</span> <span m="3030190">need</span> <span m="3030380">to</span> <span m="3030570">pass</span> <span m="3031200">straight</span> <span m="3031660">on.</span> <span m="3032760">So</span> <span m="3033000">let's</span> <span m="3033240">do</span> <span m="3033420">this.</span> <span m="3035260">So</span> <span m="3035640">over</span> <span m="3035950">here,</span> <span m="3036450">I'm in</span> <span m="3036570">011.</span> <span m="3037530">I</span> <span m="3037590">need</span> <span m="3037800">to</span> <span m="3037890">go</span> <span m="3038110">to</span> <span m="3038240">101</span> <span m="3039200">so</span> <span m="3039380">we</span> <span m="3039450">need</span> <span m="3039580">to</span> <span m="3039680">change</span> <span m="3040010">the</span> <span m="3040090">zero</span> <span m="3040370">into</span> <span m="3040710">a</span> <span m="3040790">one.</span> <span m="3042110">So</span> <span m="3043170">I</span> <span m="3043290">need</span> <span m="3043460">to</span> <span m="3043560">go</span> <span m="3043770">down.</span> <span m="3045490">I</span> <span m="3045630">need to</span> <span m="3045760">cross.</span> </p>
<p><span m="3051150">Now,</span> <span m="3053280">if</span> <span m="3053490">I</span> <span m="3053520">look</span> <span m="3053710">at</span> <span m="3053760">the</span> <span m="3053850">second</span> <span m="3054230">bit,</span> <span m="3055790">I</span> <span m="3055950">also</span> <span m="3056180">need</span> <span m="3056370">to</span> <span m="3057580">change</span> <span m="3057850">it</span> <span m="3058120">to</span> <span m="3058310">a zero</span> <span m="3058620">so</span> <span m="3058890">again,</span> <span m="3060190">I</span> <span m="3060300">need</span> <span m="3060510">to</span> <span m="3060620">cross,</span> <span m="3062550">which</span> <span m="3062730">is</span> <span m="3062860">over</span> <span m="3063170">here.</span> <span m="3065020">Now,</span> <span m="3065160">the</span> <span m="3065280">third</span> <span m="3065640">bit</span> <span m="3066300">is</span> <span m="3066500">equal</span> <span m="3066760">to</span> <span m="3066870">1</span> <span m="3069330">and</span> <span m="3069450">it's</span> <span m="3069620">the</span> <span m="3069710">same.</span> <span m="3070240">So</span> <span m="3070420">now,</span> <span m="3070630">I can</span> <span m="3070850">go</span> <span m="3071030">straight</span> <span m="3071450">ahead.</span> <span m="3071750">I</span> <span m="3071860">do</span> <span m="3071960">not</span> <span m="3072260">cross</span> <span m="3074830">and</span> <span m="3075010">I</span> <span m="3075190">end</span> <span m="3075450">up</span> <span m="3076810">at</span> <span m="3076980">this</span> <span m="3077150">output.</span> <span m="3078280">So</span> <span m="3078490">what</span> <span m="3078600">did</span> <span m="3078780">I</span> <span m="3078840">do?</span> <span m="3079380">For</span> <span m="3079680">every</span> <span m="3079920">bit</span> <span m="3080320">that</span> <span m="3080350">is</span> <span m="3080510">different,</span> <span m="3081300">I</span> <span m="3081420">cross</span> <span m="3082310">and</span> <span m="3082490">for</span> <span m="3082620">the</span> <span m="3082700">bits</span> <span m="3083060">that are</span> <span m="3083170">the</span> <span m="3083270">same,</span> <span m="3083920">I</span> <span m="3084020">go</span> <span m="3084060">straight</span> <span m="3084450">ahead.</span> <span m="3085480">So</span> <span m="3085540">this</span> <span m="3085750">is</span> <span m="3085860">how</span> <span m="3086030">I</span> <span m="3086140">can</span> <span m="3086320">route</span> <span m="3087150">packets</span> <span m="3087610">from</span> <span m="3087840">one</span> <span m="3088470">input</span> <span m="3088960">to another</span> <span m="3089090">output.</span> </p>
<p><span m="3094860">So</span> <span m="3095110">let's</span> <span m="3095540">look</span> <span m="3095910">at</span> <span m="3095980">the</span> <span m="3096060">parameters.</span> <span m="3098340">First</span> <span m="3098650">of</span> <span m="3098730">all,</span> <span m="3099030">if</span> <span m="3099180">you</span> <span m="3099280">look</span> <span m="3099560">at</span> <span m="3099800">the</span> <span m="3102240">diameter,</span> <span m="3103810">well,</span> <span m="3104040">it</span> <span m="3104160">turns</span> <span m="3104530">out</span> <span m="3104790">that</span> <span m="3104950">that's</span> <span m="3105310">approximately</span> <span m="3106480">equal</span> <span m="3106730">to</span> <span m="3108030">the</span> <span m="3108220">number</span> <span m="3108540">of</span> <span m="3108720">levels,</span> <span m="3109480">which</span> <span m="3109700">is</span> <span m="3109790">the</span> <span m="3109970">logarithm</span> <span m="3110500">of</span> <span m="3110730">n.</span> <span m="3111800">And to</span> <span m="3111880">be</span> <span m="3112030">precise,</span> <span m="3113330">it's</span> <span m="3113540">actually</span> <span m="3113820">equal</span> <span m="3114110">to</span> <span m="3114250">2</span> <span m="3115210">plus</span> <span m="3116070">the</span> <span m="3116200">logarithm</span> <span m="3116740">of</span> <span m="3116890">n.</span> <span m="3117600">So</span> <span m="3117740">that's</span> <span m="3118180">great.</span> <span m="3119760">That's</span> <span m="3119950">a</span> <span m="3120000">good</span> <span m="3120150">scaling.</span> <span m="3120650">Again,</span> <span m="3121110">it's</span> <span m="3121190">back</span> <span m="3121400">to</span> <span m="3121510">the</span> <span m="3121620">logarithm</span> <span m="3122160">of</span> <span m="3122280">n.</span> <span m="3122480">So</span> <span m="3122610">we</span> <span m="3122710">have</span> <span m="3122860">the</span> <span m="3122950">best</span> <span m="3123700">of</span> <span m="3123890">these</span> <span m="3124160">two</span> <span m="3124340">parameters.</span> </p>
<p><span m="3126410">The</span> <span m="3126520">switches</span> <span m="3126960">that</span> <span m="3127100">we</span> <span m="3127450">see</span> <span m="3128290">have</span> <span m="3128990">two</span> <span m="3129210">inputs</span> <span m="3129670">and  two</span> <span m="3129970">outputs.</span> <span m="3130790">So</span> <span m="3131060">we</span> <span m="3131220">again</span> <span m="3131570">have</span> <span m="3131930">a</span> <span m="3132180">two</span> <span m="3132420">times</span> <span m="3132860">two</span> <span m="3133090">switch.</span> <span m="3135710">The</span> <span m="3135820">number</span> <span m="3136220">of</span> <span m="3136670">switches</span> <span m="3137780">is</span> <span m="3138050">the</span> <span m="3138120">number</span> <span m="3138380">of</span> <span m="3138550">rows</span> <span m="3138960">times</span> <span m="3139250">the</span> <span m="3139320">number</span> <span m="3139550">of</span> <span m="3139690">columns.</span> <span m="3140170">The number</span> <span m="3140320">of</span> <span m="3140430">columns</span> <span m="3140760">is</span> <span m="3140890">the</span> <span m="3140990">logarithm</span> <span m="3141540">of</span> <span m="3141720">n</span> <span m="3142510">and</span> <span m="3142680">number</span> <span m="3142790">of</span> <span m="3143020">rows</span> <span m="3143400">is</span> <span m="3144520">equal</span> <span m="3144790">to</span> <span m="3144970">n.</span> <span m="3146540">And to</span> <span m="3146650">make</span> <span m="3146910">it</span> <span m="3147380">a little bit,</span> <span m="3147645">precise,</span> <span m="3147910">it's</span> <span m="3148070">1</span> <span m="3148320">plus</span> <span m="3149430">the</span> <span m="3149550">logarithm</span> <span m="3150110">of</span> <span m="3150280">n.</span> <span m="3151050">So</span> <span m="3151210">that's</span> <span m="3151520">somewhere</span> <span m="3151860">in</span> <span m="3151990">between</span> <span m="3152360">those</span> <span m="3152690">two.</span> <span m="3152910">But</span> <span m="3153120">if</span> <span m="3153250">you're</span> <span m="3153430">thinking</span> <span m="3153820">about</span> <span m="3154170">it,</span> <span m="3154320">it's</span> <span m="3154520">much</span> <span m="3154870">better</span> <span m="3155100">than</span> <span m="3155250">n</span> <span m="3155410">squared.</span> <span m="3155950">It's</span> <span m="3156200">almost</span> <span m="3156640">linear</span> <span m="3156970">except</span> <span m="3157280">for a</span> <span m="3157720">logarithmic</span> <span m="3158330">factor.</span> </p>
<p><span m="3160510">For</span> <span m="3160680">the</span> <span m="3160770">congestion--</span> <span m="3161800">and</span> <span m="3162050">we are</span> <span m="3162170">not</span> <span m="3162410">going</span> <span m="3162660">to</span> <span m="3162790">talk</span> <span m="3163080">about</span> <span m="3163400">it</span> <span m="3163830">here,</span> <span m="3164380">but</span> <span m="3164740">you</span> <span m="3164910">have</span> <span m="3165560">a</span> <span m="3165660">problem</span> <span m="3165980">set</span> <span m="3166200">assignment</span> <span m="3167080">that</span> <span m="3168300">will</span> <span m="3168750">ask</span> <span m="3169090">you</span> <span m="3169220">to</span> <span m="3169360">solve</span> <span m="3169720">this--</span> <span m="3170780">is</span> <span m="3171140">that</span> <span m="3172420">actually,</span> <span m="3173560">the</span> <span m="3173640">congestion</span> <span m="3174160">is</span> <span m="3174280">the</span> <span m="3174360">square</span> <span m="3174810">root</span> <span m="3174930">of</span> <span m="3175130">n</span> <span m="3176180">or</span> <span m="3177390">it's</span> <span m="3177610">equal</span> <span m="3177920">to</span> <span m="3178080">the</span> <span m="3178200">square</span> <span m="3178650">root</span> <span m="3179050">of</span> <span m="3179230">n</span> <span m="3179650">over</span> <span m="3179990">2,</span> <span m="3180780">depending</span> <span m="3181370">on</span> <span m="3181570">whether</span> <span m="3181880">n</span> <span m="3182300">is</span> <span m="3182720">an</span> <span m="3183280">even</span> <span m="3183580">power</span> <span m="3184300">or</span> <span m="3184500">n</span> <span m="3184690">is</span> <span m="3184870">an</span> <span m="3185060">odd</span> <span m="3185280">power.</span> <span m="3186570">Now,</span> <span m="3186710">we're</span> <span m="3186790">not</span> <span m="3186990">going</span> <span m="3187120">to</span> <span m="3187230">prove</span> <span m="3187550">that</span> <span m="3187660">here</span> <span m="3187980">because</span> <span m="3188220">we</span> <span m="3188400">want</span> <span m="3188540">to</span> <span m="3188680">step</span> <span m="3188980">forward</span> <span m="3189390">to</span> <span m="3189710">this</span> <span m="3189940">particular</span> <span m="3190300">network.</span> <span m="3190860">It's</span> <span m="3191040">very</span> <span m="3191240">exciting.</span> <span m="3192620">And</span> <span m="3193660">you</span> <span m="3193770">will</span> <span m="3193880">prove</span> <span m="3194140">this</span> <span m="3194260">in your</span> <span m="3194500">problem</span> <span m="3194840">set.</span> </p>
<p><span m="3196100">So</span> <span m="3196620">this</span> <span m="3196860">one</span> <span m="3197070">is</span> <span m="3197150">somewhere</span> <span m="3197440">in</span> <span m="3197840">between,</span> <span m="3198380">somewhere</span> <span m="3198750">in</span> <span m="3198830">between</span> <span m="3199170">these</span> <span m="3199400">two</span> <span m="3199630">extremes.</span> <span m="3200810">Now,</span> <span m="3201250">it will</span> <span m="3201450">be</span> <span m="3201620">really</span> <span m="3201930">fantastic</span> <span m="3202910">if</span> <span m="3203230">we</span> <span m="3203680">can</span> <span m="3203870">somehow</span> <span m="3205100">transform</span> <span m="3205750">this</span> <span m="3205940">network</span> <span m="3206420">with</span> <span m="3206540">a</span> <span m="3206690">trick</span> <span m="3207920">to,</span> <span m="3208580">again,</span> <span m="3209230">have</span> <span m="3209560">a</span> <span m="3209660">really</span> <span m="3210330">great</span> <span m="3210690">congestion</span> <span m="3211310">of</span> <span m="3211460">just</span> <span m="3211750">a</span> <span m="3211830">constant,</span> <span m="3212500">like</span> <span m="3214110">two</span> <span m="3214540">or</span> <span m="3214730">three</span> <span m="3215060">or</span> <span m="3215150">whatever</span> <span m="3215520">or</span> <span m="3216170">maybe</span> <span m="3216550">even</span> <span m="3216630">one.</span> </p>
<p><span m="3217910">So</span> <span m="3218700">for</span> <span m="3218820">this</span> <span m="3219000">particular</span> <span m="3219410">network,</span> <span m="3221310">in the</span> <span m="3221650">1960s,</span> <span m="3222780">Benes,</span> <span m="3223280">a Bell</span> <span m="3223570">Labs</span> <span m="3223820">researcher,</span> <span m="3224660">had</span> <span m="3224840">the</span> <span m="3224920">great</span> <span m="3225280">idea</span> <span m="3226320">to</span> <span m="3227300">use</span> <span m="3227880">a</span> <span m="3228020">butterfly</span> <span m="3228590">network</span> <span m="3229550">and</span> <span m="3229740">attach</span> <span m="3230450">to</span> <span m="3230710">it,</span> <span m="3231000">again,</span> <span m="3231490">a</span> <span m="3231750">butterfly</span> <span m="3232350">network,</span> <span m="3233240">back</span> <span m="3233460">to</span> <span m="3233590">back</span> <span m="3234100">sort</span> <span m="3234470">of.</span> </p>
<p><span m="3235680">So</span> <span m="3236950">what</span> <span m="3237130">was</span> <span m="3237310">his</span> <span m="3237500">idea?</span> <span m="3238160">His</span> <span m="3238350">idea</span> <span m="3238750">was</span> <span m="3239010">to</span> <span m="3240890">do</span> <span m="3241240">the</span> <span m="3241340">following.</span> <span m="3242910">So</span> <span m="3245620">the</span> <span m="3245720">butterfly</span> <span m="3246240">network</span> <span m="3249190">as</span> <span m="3249450">we</span> <span m="3249530">have</span> <span m="3249770">it</span> <span m="3249850">right</span> <span m="3250100">now</span> <span m="3250500">is</span> <span m="3250690">this</span> <span m="3250880">particular</span> <span m="3251370">part</span> <span m="3251750">over</span> <span m="3252020">here.</span> <span m="3254080">And</span> <span m="3254170">the</span> <span m="3254290">idea</span> <span m="3254710">is</span> <span m="3256430">now</span> <span m="3257030">to</span> <span m="3257200">start</span> <span m="3257650">up</span> <span m="3258310">mixing</span> <span m="3259040">all</span> <span m="3259280">those</span> <span m="3259610">outputs</span> <span m="3260300">that</span> <span m="3260450">we</span> <span m="3260600">got</span> <span m="3260840">here</span> <span m="3261320">together</span> <span m="3261910">again</span> <span m="3262450">using</span> <span m="3262990">a</span> <span m="3263070">similar</span> <span m="3263520">rule.</span> </p>
<p><span m="3264550">So</span> <span m="3264770">what</span> <span m="3264880">do</span> <span m="3264980">we</span> <span m="3265100">do?</span> <span m="3265990">We</span> <span m="3266200">are</span> <span m="3266270">going</span> <span m="3266640">to</span> <span m="3268590">essentially</span> <span m="3275630">repeat</span> <span m="3276270">this</span> <span m="3276480">particular</span> <span m="3276930">structure</span> <span m="3278030">on</span> <span m="3278270">this</span> <span m="3278560">side.</span> <span m="3281030">So</span> <span m="3281250">how</span> <span m="3281360">do</span> <span m="3281470">we</span> <span m="3281570">do</span> <span m="3281770">it?</span> <span m="3283090">Well,</span> <span m="3283410">we</span> <span m="3283720">go</span> <span m="3284130">either</span> <span m="3284380">straightforward</span> <span m="3288850">or</span> <span m="3289560">we</span> <span m="3289650">start</span> <span m="3290060">to</span> <span m="3290170">mix</span> <span m="3290420">them</span> <span m="3290610">again.</span> <span m="3291070">So</span> <span m="3291870">it's</span> <span m="3292140">like</span> <span m="3292430">this</span> <span m="3292710">output,</span> <span m="3294090">this</span> <span m="3294380">particular</span> <span m="3294740">switch,</span> <span m="3295150">can</span> <span m="3295330">either</span> <span m="3295890">go</span> <span m="3296070">straight</span> <span m="3296530">ahead</span> <span m="3296810">or can</span> <span m="3297190">cross</span> <span m="3297960">to</span> <span m="3298100">the</span> <span m="3298220">lower</span> <span m="3300050">part</span> <span m="3300360">over</span> <span m="3300660">here.</span> <span m="3301940">It</span> <span m="3302130">goes</span> <span m="3302390">over</span> <span m="3302640">here</span> <span m="3302900">and</span> <span m="3303020">this</span> <span m="3303280">one goes</span> <span m="3303540">over.</span> </p>
<p><span m="3303860">So</span> <span m="3304040">as</span> <span m="3304280">you</span> <span m="3304380">can</span> <span m="3304530">see,</span> <span m="3304875">we have</span> <span m="3305690">repeated</span> <span m="3306220">this</span> <span m="3306460">part.</span> <span m="3308240">It's</span> <span m="3308410">exactly</span> <span m="3308780">the</span> <span m="3308900">same</span> <span m="3309250">as</span> <span m="3309380">this</span> <span m="3309570">structure</span> <span m="3310000">over</span> <span m="3310300">here.</span> <span m="3312040">We'll</span> <span m="3312170">do</span> <span m="3312280">the</span> <span m="3312460">same</span> <span m="3315750">for</span> <span m="3316990">this</span> <span m="3317310">part.</span> <span m="3319820">So</span> <span m="3319920">we</span> <span m="3320040">can</span> <span m="3320210">either</span> <span m="3320490">cross</span> <span m="3323530">or</span> <span m="3323660">we</span> <span m="3323760">can</span> <span m="3323910">go</span> <span m="3324060">straight</span> <span m="3324430">ahead.</span> <span m="3324910">Oh,</span> <span m="3325490">we</span> <span m="3325620">also</span> <span m="3325890">have,</span> <span m="3326150">of</span> <span m="3326180">course,</span> <span m="3326450">that</span> <span m="3326570">these</span> <span m="3326760">switches</span> <span m="3327390">can</span> <span m="3327540">go</span> <span m="3327770">straight</span> <span m="3328150">ahead</span> <span m="3328800">or</span> <span m="3329020">can</span> <span m="3329130">cross</span> <span m="3329720">to</span> <span m="3330040">the</span> <span m="3330150">top.</span> <span m="3331470">I</span> <span m="3331590">forgot</span> <span m="3331920">about</span> <span m="3332200">that.</span> <span m="3333400">So</span> <span m="3333640">we</span> <span m="3333750">have</span> <span m="3333960">this--</span> <span m="3335140">oops--</span> <span m="3335580">as</span> <span m="3335830">well.</span> <span m="3337080">So</span> <span m="3337180">as</span> <span m="3337260">you</span> <span m="3337370">can</span> <span m="3337520">see,</span> <span m="3338150">this</span> <span m="3338360">particular</span> <span m="3338800">structure</span> <span m="3339320">repeats</span> <span m="3339770">itself</span> <span m="3340180">again</span> <span m="3341350">and</span> <span m="3341500">we</span> <span m="3341590">slowly</span> <span m="3341960">start</span> <span m="3342300">to</span> <span m="3342380">build</span> <span m="3342690">up</span> <span m="3342860">in</span> <span m="3342980">mixing</span> <span m="3343410">all</span> <span m="3343610">the</span> <span m="3344520">outputs</span> <span m="3345090">again</span> <span m="3345820">or</span> <span m="3346090">the</span> <span m="3346220">possibility,</span> <span m="3346525">at</span> <span m="3346830">least,</span> <span m="3347070">to</span> <span m="3347220">route</span> <span m="3347510">them</span> <span m="3347700">to</span> <span m="3347960">any</span> <span m="3348500">other</span> <span m="3353310">row.</span> </p>
<p><span m="3354510">So</span> <span m="3354710">how</span> <span m="3354830">do</span> <span m="3354940">we</span> <span m="3355080">do</span> <span m="3355210">this?</span> <span m="3355810">Well,</span> <span m="3355940">we</span> <span m="3356070">continue</span> <span m="3356600">this</span> <span m="3356770">particular</span> <span m="3357180">structure</span> <span m="3357630">now</span> <span m="3357920">over</span> <span m="3358210">here.</span> <span m="3358500">So</span> <span m="3359680">all</span> <span m="3360000">these</span> <span m="3360280">can</span> <span m="3360460">either</span> <span m="3360770">go</span> <span m="3361020">straight</span> <span m="3361480">ahead.</span> <span m="3367560">That's</span> <span m="3367780">a</span> <span m="3367840">possibility.</span> <span m="3371570">Or</span> <span m="3372380">they</span> <span m="3372770">can</span> <span m="3373010">go</span> <span m="3373290">all</span> <span m="3374940">down.</span> <span m="3378690">So</span> <span m="3379100">this</span> <span m="3379380">switch</span> <span m="3379600">can</span> <span m="3379810">either</span> <span m="3380020">go</span> <span m="3380170">straight</span> <span m="3380510">ahead</span> <span m="3380850">or</span> <span m="3381040">can</span> <span m="3381850">go</span> <span m="3382020">to</span> <span m="3382160">the</span> <span m="3382290">lower</span> <span m="3382650">half.</span> <span m="3384790">And</span> <span m="3384950">for</span> <span m="3385030">these,</span> <span m="3385240">we</span> <span m="3385400">have</span> <span m="3385610">a</span> <span m="3385670">similar</span> <span m="3386070">structure.</span> <span m="3386610">We</span> <span m="3386810">can either</span> <span m="3386900">go</span> <span m="3387060">straight</span> <span m="3387430">ahead</span> <span m="3390840">or</span> <span m="3391280">such</span> <span m="3391550">a</span> <span m="3391610">switch</span> <span m="3392160">can</span> <span m="3392330">cross</span> <span m="3392650">over</span> <span m="3393340">to</span> <span m="3393450">the</span> <span m="3393620">top</span> <span m="3395870">over</span> <span m="3396160">here.</span> </p>
<p><span m="3402180">So</span> <span m="3402380">that's</span> <span m="3402630">this.</span> <span m="3403480">So</span> <span m="3403660">this</span> <span m="3403870">is</span> <span m="3404000">Benes</span> <span m="3404360">network</span> <span m="3405090">and</span> <span m="3405230">then</span> <span m="3405410">over</span> <span m="3405670">here,</span> <span m="3405870">of</span> <span m="3406000">course,</span> <span m="3406300">we</span> <span m="3406420">have</span> <span m="3408220">the</span> <span m="3408410">outputs,</span> <span m="3409410">zero,</span> <span m="3413810">one,</span> <span m="3415230">and</span> <span m="3415410">all</span> <span m="3415570">the</span> <span m="3415660">way</span> <span m="3415810">down</span> <span m="3418790">to</span> <span m="3419480">seven.</span> <span m="3431340">So</span> <span m="3431550">as</span> <span m="3431660">you</span> <span m="3431810">can</span> <span m="3432010">see</span> <span m="3432380">over</span> <span m="3432640">here,</span> <span m="3433260">the</span> <span m="3433390">structure</span> <span m="3434740">again</span> <span m="3435060">has</span> <span m="3435260">a recursive</span> <span m="3435870">nature</span> <span m="3436280">to</span> <span m="3436540">it.</span> </p>
<p><span m="3437980">You</span> <span m="3438120">can</span> <span m="3438320">see</span> <span m="3438810">that</span> <span m="3440120">this</span> <span m="3440370">big</span> <span m="3441380">Benes</span> <span m="3441860">network</span> <span m="3442590">over</span> <span m="3442910">here</span> <span m="3444340">consists</span> <span m="3445560">of</span> <span m="3446080">two</span> <span m="3446310">smaller</span> <span m="3446740">ones</span> <span m="3447260">that</span> <span m="3447350">are</span> <span m="3447510">right</span> <span m="3447730">here</span> <span m="3447920">in</span> <span m="3447990">the</span> <span m="3448050">middle,</span> <span m="3449350">this</span> <span m="3449680">one</span> <span m="3450210">that</span> <span m="3450390">goes</span> <span m="3450730">all</span> <span m="3450930">the</span> <span m="3451040">way</span> <span m="3451280">up</span> <span m="3451530">to</span> <span m="3451670">here--</span> <span m="3452150">so</span> <span m="3452210">maybe</span> <span m="3452350">I</span> <span m="3452450">should</span> <span m="3453560">put</span> <span m="3455150">a</span> <span m="3455290">color</span> <span m="3455540">boundary</span> <span m="3456300">around</span> <span m="3456700">it.</span> <span m="3457290">Let</span> <span m="3457450">me</span> <span m="3457710">check</span> <span m="3459250">I</span> <span m="3459410">want</span> <span m="3459640">to</span> <span m="3459750">do</span> <span m="3459970">this--</span> <span m="3463990">right.</span> <span m="3464280">So</span> <span m="3465570">this</span> <span m="3465760">particular</span> <span m="3466230">part,</span> <span m="3471020">is</span> <span m="3471250">again</span> <span m="3472150">a</span> <span m="3472410">Benes</span> <span m="3473090">network</span> <span m="3474370">and</span> <span m="3476620">the</span> <span m="3476790">top</span> <span m="3477030">part</span> <span m="3477600">in</span> <span m="3477740">the</span> <span m="3477820">same</span> <span m="3478090">picture,</span> <span m="3478810">the</span> <span m="3478930">top</span> <span m="3479490">subnetwork</span> <span m="3482300">is</span> <span m="3482520">also</span> <span m="3483010">a</span> <span m="3483070">Benes</span> <span m="3483460">network,</span> <span m="3484800">this</span> <span m="3485300">part.</span> </p>
<p><span m="3493300">And</span> <span m="3493560">if</span> <span m="3493680">you</span> <span m="3493850">look</span> <span m="3494130">within</span> <span m="3494510">those,</span> <span m="3495420">we</span> <span m="3495550">again</span> <span m="3495880">see</span> <span m="3497000">a</span> <span m="3497160">top</span> <span m="3497520">part</span> <span m="3499170">and</span> <span m="3499570">a</span> <span m="3499640">bottom</span> <span m="3500000">part.</span> <span m="3500600">And</span> <span m="3500750">over</span> <span m="3501000">here,</span> <span m="3501260">we</span> <span m="3501340">see</span> <span m="3501600">a</span> <span m="3501710">top</span> <span m="3501970">part</span> <span m="3503370">and</span> <span m="3503610">also</span> <span m="3503920">a</span> <span m="3503980">bottom</span> <span m="3504300">part.</span> <span m="3505830">So</span> <span m="3508290">you</span> <span m="3508410">see</span> <span m="3508630">this</span> <span m="3509350">recursive</span> <span m="3509860">nature</span> <span m="3510250">again</span> <span m="3510690">reappearing.</span> <span m="3511500">It</span> <span m="3511700">turns</span> <span m="3512080">out</span> <span m="3512870">that</span> <span m="3513010">with</span> <span m="3513170">this</span> <span m="3513480">trick,</span> <span m="3513930">we</span> <span m="3514060">can</span> <span m="3514300">completely</span> <span m="3514900">eliminate</span> <span m="3515460">congestion</span> <span m="3516620">and</span> <span m="3516780">we</span> <span m="3516870">can</span> <span m="3517060">get</span> <span m="3517280">it</span> <span m="3517590">to</span> <span m="3518180">only</span> <span m="3518860">one,</span> <span m="3519370">which</span> <span m="3519580">is</span> <span m="3519690">really</span> <span m="3519880">surprising.</span> <span m="3520600">And</span> <span m="3520760">that</span> <span m="3520920">what</span> <span m="3521040">we're</span> <span m="3521160">going</span> <span m="3521380">to</span> <span m="3521500">prove</span> <span m="3521830">here.</span> <span m="3522800">So</span> <span m="3522830">this</span> <span m="3522980">is</span> <span m="3523120">a</span> <span m="3523170">great</span> <span m="3524310">invention</span> <span m="3524870">at</span> <span m="3524970">the</span> <span m="3525050">time.</span> <span m="3525690">It's</span> <span m="3525840">really,</span> <span m="3526300">really</span> <span m="3526420">beautiful.</span> </p>
<p><span m="3528870">So</span> <span m="3529060">let</span> <span m="3529170">me</span> <span m="3529300">put</span> <span m="3529470">in</span> <span m="3529610">the</span> <span m="3529750">other</span> <span m="3529900">parameters.</span> <span m="3531880">So</span> <span m="3532560">they</span> <span m="3532670">stay</span> <span m="3532930">approximately</span> <span m="3533540">the</span> <span m="3533670">same</span> <span m="3534080">up</span> <span m="3534300">to</span> <span m="3535410">that the</span> <span m="3535750">diameter</span> <span m="3536250">is</span> <span m="3536430">about</span> <span m="3536770">twice</span> <span m="3537100">as</span> <span m="3537240">large</span> <span m="3537670">because</span> <span m="3538040">we</span> <span m="3538180">added</span> <span m="3538820">another</span> <span m="3539510">sort</span> <span m="3539870">of</span> <span m="3539980">whole</span> <span m="3540360">butterfly</span> <span m="3540850">structure</span> <span m="3541250">to</span> <span m="3541500">it.</span> <span m="3542520">The</span> <span m="3542620">switch</span> <span m="3543990">size</span> <span m="3544440">stays</span> <span m="3544690">the</span> <span m="3544800">same.</span> <span m="3545960">We,</span> <span m="3546310">again,</span> <span m="3546660">have</span> <span m="3546880">about</span> <span m="3547240">two</span> <span m="3547400">times</span> <span m="3547740">more</span> <span m="3547950">switches</span> <span m="3551190">so</span> <span m="3552150">they</span> <span m="3552310">sort</span> <span m="3552590">of</span> <span m="3552700">stay</span> <span m="3553220">about</span> <span m="3553560">the</span> <span m="3553650">same</span> <span m="3553960">up</span> <span m="3554100">to</span> <span m="3554270">a</span> <span m="3554300">linear</span> <span m="3555320">factor,</span> <span m="3556080">like</span> <span m="3556340">a</span> <span m="3556930">constant</span> <span m="3557280">factor.</span> <span m="3558380">And</span> <span m="3559230">the</span> <span m="3559320">congestion,</span> <span m="3559870">however,</span> <span m="3560510">completely</span> <span m="3561010">dropped</span> <span m="3561280">down</span> <span m="3561500">to</span> <span m="3561650">one.</span> </p>
<p><span m="3562210">So</span> <span m="3562350">that's</span> <span m="3562520">what</span> <span m="3562620">we're going</span> <span m="3562940">to</span> <span m="3563060">prove</span> <span m="3563460">now.</span> <span m="3564490">And</span> <span m="3565030">in order</span> <span m="3565285">to</span> <span m="3565540">get</span> <span m="3565780">some</span> <span m="3566370">intuition,</span> <span m="3567610">well,</span> <span m="3567740">let</span> <span m="3567830">me</span> <span m="3567930">first</span> <span m="3568360">write</span> <span m="3568600">down</span> <span m="3568780">the</span> <span m="3568850">theorem.</span> <span m="3572330">Actually,</span> <span m="3572690">let</span> <span m="3572810">me</span> <span m="3572950">put</span> <span m="3573170">this</span> <span m="3576410">over</span> <span m="3576700">here.</span> <span m="3583840">So</span> <span m="3584620">in</span> <span m="3584770">order</span> <span m="3584910">to</span> <span m="3585020">get</span> <span m="3585200">some</span> <span m="3585390">insight</span> <span m="3586000">into</span> <span m="3586330">this,</span> <span m="3594530">we are</span> <span m="3594830">going to</span> <span m="3594940">use</span> <span m="3595140">this</span> <span m="3595380">recursive</span> <span m="3595890">nature.</span> <span m="3596410">So</span> <span m="3596600">we're going</span> <span m="3596880">to use</span> <span m="3597130">induction</span> <span m="3597760">and</span> <span m="3597830">we're going</span> <span m="3598140">to</span> <span m="3598270">say,</span> <span m="3598610">oh,</span> <span m="3601630">for</span> <span m="3601890">any</span> <span m="3602240">permutation,</span> <span m="3603100">I</span> <span m="3603250">can</span> <span m="3603450">find</span> <span m="3603860">really</span> <span m="3604210">good</span> <span m="3604470">routing</span> <span m="3605210">for</span> <span m="3605555">say,</span> <span m="3605900">this</span> <span m="3606430">red</span> <span m="3606760">subnetwork</span> <span m="3607910">and</span> <span m="3608070">for</span> <span m="3608210">this</span> <span m="3608450">blue</span> <span m="3609040">subnetwork.</span> <span m="3609870">So</span> <span m="3610030">I</span> <span m="3610100">know</span> <span m="3610320">that.</span> </p>
<p><span m="3611580">So</span> <span m="3611780">what</span> <span m="3611920">I</span> <span m="3611970">need</span> <span m="3612210">to</span> <span m="3612340">do</span> <span m="3612930">is,</span> <span m="3615430">if</span> <span m="3615660">I</span> <span m="3615750">have</span> <span m="3615890">my</span> <span m="3615980">bigger</span> <span m="3616930">Benes</span> <span m="3617470">network,</span> <span m="3617960">like</span> <span m="3618110">this</span> <span m="3618310">one,</span> <span m="3619110">I</span> <span m="3619280">would</span> <span m="3619480">need</span> <span m="3619680">to</span> <span m="3619810">somehow</span> <span m="3621010">map</span> <span m="3621450">these</span> <span m="3621680">inputs--</span> <span m="3624360">I</span> <span m="3624460">need</span> <span m="3624640">to</span> <span m="3624790">route</span> <span m="3625100">them</span> <span m="3625330">to</span> <span m="3625500">either</span> <span m="3626150">the</span> <span m="3626300">top</span> <span m="3626590">and</span> <span m="3626750">the</span> <span m="3626820">bottom</span> <span m="3627470">subnetwork,</span> <span m="3628590">one</span> <span m="3628830">of</span> <span m="3628940">the</span> <span m="3629080">two,</span> <span m="3629360">in</span> <span m="3629440">such</span> <span m="3629800">a</span> <span m="3629890">way</span> <span m="3630510">that</span> <span m="3630680">there</span> <span m="3630770">will</span> <span m="3631770">be</span> <span m="3632230">absolutely</span> <span m="3632840">no</span> <span m="3633050">congestion,</span> <span m="3633620">because</span> <span m="3633860">we</span> <span m="3634000">want</span> <span m="3634140">to</span> <span m="3634240">keep</span> <span m="3634490">this</span> <span m="3634670">one.</span> <span m="3635080">So</span> <span m="3635600">a</span> <span m="3635700">switch</span> <span m="3636080">should</span> <span m="3636540">only</span> <span m="3636880">see</span> <span m="3637150">one</span> <span m="3637390">packet</span> <span m="3637770">coming</span> <span m="3638080">in.</span> <span m="3639300">So</span> <span m="3639430">that</span> <span m="3639630">means,</span> <span m="3639990">for</span> <span m="3640180">example--</span> <span m="3641400">and</span> <span m="3641590">we'll</span> <span m="3641780">come</span> <span m="3641930">back</span> <span m="3642110">to</span> <span m="3642240">that--</span> <span m="3642460">that</span> <span m="3642780">for</span> <span m="3642960">example,</span> <span m="3643110">for this</span> <span m="3643350">switch,</span> <span m="3644090">it</span> <span m="3644190">should</span> <span m="3644390">not</span> <span m="3644610">receive</span> <span m="3644960">a</span> <span m="3645070">packet</span> <span m="3645410">from</span> <span m="3645560">both</span> <span m="3646060">this</span> <span m="3646300">input</span> <span m="3647260">and</span> <span m="3647480">from</span> <span m="3647640">this</span> <span m="3647850">input.</span> </p>
<p><span m="3649080">So</span> <span m="3649230">the</span> <span m="3649320">intuition</span> <span m="3649810">that</span> <span m="3649950">we are</span> <span m="3650190">going</span> <span m="3650460">to</span> <span m="3650610">create</span> <span m="3651530">is</span> <span m="3652030">we're going</span> <span m="3652530">to</span> <span m="3656190">list</span> <span m="3656540">our</span> <span m="3656780">constraints,</span> <span m="3657610">the</span> <span m="3657700">constraints</span> <span m="3658510">that</span> <span m="3658660">we</span> <span m="3658780">need</span> <span m="3658940">to</span> <span m="3659020">satisfy,</span> <span m="3659840">like</span> <span m="3660810">the</span> <span m="3660900">zero</span> <span m="3661330">and</span> <span m="3661470">the</span> <span m="3661550">fourth</span> <span m="3661920">input</span> <span m="3662930">should</span> <span m="3663170">not</span> <span m="3663490">both</span> <span m="3664040">be</span> <span m="3664190">mapped</span> <span m="3665060">to</span> <span m="3665220">this</span> <span m="3665590">top</span> <span m="3665960">subnetwork</span> <span m="3667330">and</span> <span m="3667540">so</span> <span m="3667720">on.</span> <span m="3667950">So</span> <span m="3668020">we will</span> <span m="3668200">get</span> <span m="3668370">into</span> <span m="3668630">that</span> <span m="3669370">and</span> <span m="3669520">then</span> <span m="3669960">we</span> <span m="3670090">will</span> <span m="3671450">gain</span> <span m="3671770">a</span> <span m="3671830">lot</span> <span m="3671960">of</span> <span m="3672060">intuition</span> <span m="3672620">on</span> <span m="3672740">how</span> <span m="3672940">to</span> <span m="3673430">solve this.</span> </p>
<p><span m="3675100">So</span> <span m="3675250">what's</span> <span m="3675450">the</span> <span m="3675530">theorem?</span> <span m="3677970">So</span> <span m="3678160">the</span> <span m="3678270">theorem</span> <span m="3678720">is</span> <span m="3679130">that</span> <span m="3679240">the</span> <span m="3680311">congestion</span> <span m="3685030">of</span> <span m="3686910">the</span> <span m="3687270">n-input</span> <span m="3690180">Benes</span> <span m="3692970">network</span> <span m="3697490">is</span> <span m="3697680">actually</span> <span m="3697960">equal</span> <span m="3698250">to</span> <span m="3698390">1.</span> <span m="3701110">And</span> <span m="3701290">we</span> <span m="3701460">will</span> <span m="3701570">prove</span> <span m="3701940">this</span> <span m="3702260">for</span> <span m="3703720">n</span> <span m="3704470">equal</span> <span m="3704940">to</span> <span m="3705170">a</span> <span m="3705230">power</span> <span m="3705590">of</span> <span m="3705750">2.</span> <span m="3706320">We</span> <span m="3706460">have</span> <span m="3706570">assumed</span> <span m="3706870">that</span> <span m="3707860">at</span> <span m="3707990">the</span> <span m="3708070">start</span> <span m="3710260">that we</span> <span m="3710390">had</span> <span m="3710560">with</span> <span m="3710690">all</span> <span m="3710820">the</span> <span m="3710950">other</span> <span m="3711140">networks,</span> <span m="3711570">as</span> <span m="3711710">well.</span> <span m="3714280">And</span> <span m="3714460">in</span> <span m="3714540">this</span> <span m="3714810">case,</span> <span m="3716460">we</span> <span m="3716630">will</span> <span m="3717180">use</span> <span m="3717500">induction</span> <span m="3718870">on</span> <span m="3719160">a.</span> <span m="3720340">So</span> <span m="3720910">that's</span> <span m="3723108">the</span> <span m="3723550">method</span> <span m="3724030">that</span> <span m="3724060">we</span> <span m="3724180">will</span> <span m="3724280">do</span> <span m="3724520">because</span> <span m="3725190">that's</span> <span m="3725410">also</span> <span m="3725610">the</span> <span m="3725640">recursive</span> <span m="3726250">structure</span> <span m="3726790">of</span> <span m="3726970">the</span> <span m="3727110">Benes</span> <span m="3727470">network</span> <span m="3727850">itself.</span> <span m="3729150">So</span> <span m="3729320">we</span> <span m="3729420">will</span> <span m="3729580">use</span> <span m="3729830">induction</span> <span m="3734080">on</span> <span m="3734470">a</span> <span m="3736390">and</span> <span m="3737740">we</span> <span m="3737930">are going</span> <span m="3738180">to</span> <span m="3738310">define</span> <span m="3738990">the</span> <span m="3739130">induction</span> <span m="3739410">hypothesis</span> <span m="3740590">simply</span> <span m="3741150">as,</span> <span m="3742070">"The</span> <span m="3742190">theorem</span> <span m="3742650">is</span> <span m="3742800">true</span> <span m="3743030">for</span> <span m="3743250">a."</span> </p>
<p><span m="3754210">Now,</span> <span m="3754340">let</span> <span m="3754530">us</span> <span m="3754750">do</span> <span m="3754900">the</span> <span m="3755020">base</span> <span m="3755330">case.</span> <span m="3755570">We always</span> <span m="3755870">start</span> <span m="3756140">with</span> <span m="3756240">the</span> <span m="3756330">base</span> <span m="3756680">case</span> <span m="3757840">and</span> <span m="3759020">that</span> <span m="3759180">should</span> <span m="3759340">be</span> <span m="3759480">pretty</span> <span m="3759870">easy</span> <span m="3760490">because</span> <span m="3760840">this</span> <span m="3761010">is</span> <span m="3761140">the</span> <span m="3761220">most</span> <span m="3761580">basic</span> <span m="3762760">Benes</span> <span m="3763200">network.</span> <span m="3764240">So</span> <span m="3764600">n</span> <span m="3765030">equals</span> <span m="3765550">2</span> <span m="3765750">to</span> <span m="3765870">the</span> <span m="3765980">power</span> <span m="3766320">of 1.</span> <span m="3768630">We</span> <span m="3768780">essentially</span> <span m="3769410">have</span> <span m="3770570">two</span> <span m="3770800">inputs,</span> <span m="3771960">an</span> <span m="3772130">input</span> <span m="3772670">zero</span> <span m="3774516">and an</span> <span m="3774920">input</span> <span m="3775520">one.</span> <span m="3777210">They are</span> <span m="3777440">connected</span> <span m="3777890">to</span> <span m="3778060">these</span> <span m="3778290">switches</span> <span m="3779080">over</span> <span m="3779390">here</span> <span m="3779620">that</span> <span m="3779810">can</span> <span m="3780020">either</span> <span m="3780280">forward</span> <span m="3780900">them</span> <span m="3782450">or</span> <span m="3782730">can</span> <span m="3782920">cross</span> <span m="3783330">them</span> <span m="3783600">over</span> <span m="3785010">and</span> <span m="3785240">then</span> <span m="3785390">they</span> <span m="3785480">go</span> <span m="3785700">directly</span> <span m="3786330">to</span> <span m="3786990">the</span> <span m="3787180">output.</span> </p>
<p><span m="3788850">Notice</span> <span m="3789250">that in</span> <span m="3789420">this</span> <span m="3789640">case,</span> <span m="3789870">we</span> <span m="3789940">just</span> <span m="3790250">have</span> <span m="3790570">the</span> <span m="3790820">most</span> <span m="3791220">elementary</span> <span m="3791800">butterfly</span> <span m="3792280">network.</span> <span m="3792800">It's the</span> <span m="3793085">same.</span> <span m="3794780">So</span> <span m="3795050">we have</span> <span m="3795340">output</span> <span m="3796240">zero</span> <span m="3796870">and</span> <span m="3797180">output</span> <span m="3798460">one.</span> <span m="3799160">So</span> <span m="3799310">this</span> <span m="3799490">corresponds</span> <span m="3800010">in</span> <span m="3800110">this</span> <span m="3800310">picture</span> <span m="3801270">to</span> <span m="3801430">these</span> <span m="3802720">little</span> <span m="3802990">small</span> <span m="3803420">things</span> <span m="3803770">over</span> <span m="3804070">here,</span> <span m="3805090">this</span> <span m="3805350">one</span> <span m="3805900">and</span> <span m="3806080">this</span> <span m="3806320">one</span> <span m="3807070">and</span> <span m="3807220">this</span> <span m="3807390">one</span> <span m="3807540">over</span> <span m="3807790">here</span> <span m="3808620">and</span> <span m="3808800">the</span> <span m="3808870">fourth</span> <span m="3809145">one</span> <span m="3809420">over</span> <span m="3809660">here.</span> <span m="3812440">So</span> <span m="3812550">now,</span> <span m="3812740">let's</span> <span m="3813030">take</span> <span m="3813520">any</span> <span m="3813770">permutation.</span> <span m="3814630">We</span> <span m="3814740">want</span> <span m="3814890">to</span> <span m="3814980">show</span> <span m="3815230">that</span> <span m="3815370">we</span> <span m="3815460">can</span> <span m="3815700">route</span> <span m="3816010">it</span> <span m="3816130">in</span> <span m="3816230">such</span> <span m="3816530">a</span> <span m="3816600">way</span> <span m="3817300">that</span> <span m="3817760">there's</span> <span m="3817950">only</span> <span m="3818530">a</span> <span m="3818600">congestion</span> <span m="3819130">of</span> <span m="3819250">one.</span> <span m="3820910">So</span> <span m="3821110">let's</span> <span m="3821310">do</span> <span m="3821480">this.</span> </p>
<p><span m="3831970">So</span> <span m="3832110">there are</span> <span m="3832310">essentially</span> <span m="3832740">only</span> <span m="3832970">two</span> <span m="3833160">permutations.</span> <span m="3834620">Either</span> <span m="3835350">zero</span> <span m="3835750">is</span> <span m="3835880">mapped</span> <span m="3836110">to</span> <span m="3836180">zero</span> <span m="3836650">and</span> <span m="3836780">one</span> <span m="3836960">is</span> <span m="3837070">mapped</span> <span m="3837300">to</span> <span m="3837410">one</span> <span m="3837960">or</span> <span m="3838100">zero</span> <span m="3838490">is</span> <span m="3838550">mapped</span> <span m="3838840">to</span> <span m="3838990">one</span> <span m="3839650">and</span> <span m="3840080">one</span> <span m="3840380">is</span> <span m="3840530">mapped</span> <span m="3840750">to</span> <span m="3840830">zero.</span> <span m="3844280">So</span> <span m="3844620">in</span> <span m="3844740">both</span> <span m="3845060">cases,</span> <span m="3845990">we</span> <span m="3846140">can</span> <span m="3846300">just</span> <span m="3846630">route</span> <span m="3846910">them</span> <span m="3847690">through</span> <span m="3849250">their</span> <span m="3849450">own</span> <span m="3849680">switches.</span> <span m="3851890">So</span> <span m="3852230">we</span> <span m="3852360">have</span> <span m="3852820">that</span> <span m="3853630">either</span> <span m="3854350">pi of</span> <span m="3854740">0</span> <span m="3855130">equals</span> <span m="3855520">0</span> <span m="3856000">and</span> <span m="3856290">pi</span> <span m="3857170">of</span> <span m="3857420">1</span> <span m="3857870">equals</span> <span m="3858430">1,</span> <span m="3859500">in</span> <span m="3859650">which</span> <span m="3859900">case</span> <span m="3860580">we</span> <span m="3860690">just</span> <span m="3861080">direct</span> <span m="3861420">them</span> <span m="3861590">straight</span> <span m="3862000">through</span> <span m="3863460">and</span> <span m="3863630">we</span> <span m="3863710">go</span> <span m="3863900">straight</span> <span m="3864280">through</span> <span m="3865080">and</span> <span m="3865260">every</span> <span m="3865490">switch</span> <span m="3866140">only</span> <span m="3866410">sees</span> <span m="3866700">a packet</span> <span m="3867110">once.</span> <span m="3867930">So</span> <span m="3868060">for</span> <span m="3868210">this</span> <span m="3868410">particular</span> <span m="3869200">permutation,</span> <span m="3870310">we</span> <span m="3870450">have a</span> <span m="3870730">congestion</span> <span m="3871290">of</span> <span m="3871400">one.</span> </p>
<p><span m="3872810">Now,</span> <span m="3872940">the</span> <span m="3873080">other</span> <span m="3873950">permutation</span> <span m="3874640">that</span> <span m="3874790">we</span> <span m="3874900">can</span> <span m="3875110">have</span> <span m="3876160">is</span> <span m="3876390">if</span> <span m="3876600">zero</span> <span m="3876980">is</span> <span m="3877120">mapped</span> <span m="3877330">to</span> <span m="3877470">one</span> <span m="3878550">and</span> <span m="3878770">if</span> <span m="3878960">one</span> <span m="3879320">is</span> <span m="3879570">mapped</span> <span m="3879835">to</span> <span m="3880100">zero.</span> <span m="3880660">Well,</span> <span m="3881080">in</span> <span m="3881270">that</span> <span m="3881490">case,</span> <span m="3883150">we</span> <span m="3883260">just</span> <span m="3883710">route</span> <span m="3884110">this</span> <span m="3885310">cross</span> <span m="3885710">over</span> <span m="3886400">to</span> <span m="3886680">the</span> <span m="3886800">bottom</span> <span m="3887760">row</span> <span m="3888400">and</span> <span m="3888540">here</span> <span m="3888750">we</span> <span m="3888850">go</span> <span m="3889200">from</span> <span m="3889550">this</span> <span m="3889740">switch</span> <span m="3890070">to</span> <span m="3890170">the</span> <span m="3890780">top</span> <span m="3891020">row.</span> <span m="3891650">Again,</span> <span m="3892240">every</span> <span m="3892520">switch</span> <span m="3893300">only</span> <span m="3893590">sees</span> <span m="3893860">a packet</span> <span m="3894330">once.</span> <span m="3895490">So</span> <span m="3895600">in</span> <span m="3895670">this</span> <span m="3895910">case,</span> <span m="3896860">in</span> <span m="3896960">the</span> <span m="3897040">base</span> <span m="3897350">case,</span> <span m="3897850">we</span> <span m="3898000">are</span> <span m="3898050">done.</span> <span m="3898460">We are</span> <span m="3898630">happy.</span> <span m="3899330">We</span> <span m="3899550">have</span> <span m="3899830">shown</span> <span m="3900220">that</span> <span m="3900570">the</span> <span m="3900670">congestion</span> <span m="3901170">is</span> <span m="3901270">equal</span> <span m="3901550">to</span> <span m="3901680">one.</span> </p>
<p><span m="3902871">So</span> <span m="3903270">now,</span> <span m="3903500">it</span> <span m="3903600">gets</span> <span m="3903830">to</span> <span m="3906050">the</span> <span m="3906130">harder</span> <span m="3906510">part</span> <span m="3907060">because</span> <span m="3907810">for the</span> <span m="3908120">inductive</span> <span m="3908470">step,</span> <span m="3910500">we</span> <span m="3910640">are</span> <span m="3910680">going</span> <span m="3910960">to</span> <span m="3911110">assume,</span> <span m="3911980">of</span> <span m="3912140">course,</span> <span m="3915280">that</span> <span m="3915440">it</span> <span m="3915560">holds</span> <span m="3915850">true</span> <span m="3916120">for a</span> <span m="3916370">smaller</span> <span m="3918080">Benes</span> <span m="3918620">network.</span> <span m="3919200">So</span> <span m="3919400">we</span> <span m="3919520">assume</span> <span m="3921770">that</span> <span m="3922590">P</span> <span m="3922985">a</span> <span m="3924170">is</span> <span m="3924390">true</span> <span m="3925890">and</span> <span m="3928870">well,</span> <span m="3929090">let's</span> <span m="3929330">try</span> <span m="3929540">to</span> <span m="3930210">gain</span> <span m="3930350">some</span> <span m="3930520">insight</span> <span m="3931020">here.</span> <span m="3932980">So</span> <span m="3933990">we</span> <span m="3934190">know</span> <span m="3936140">from</span> <span m="3936420">our</span> <span m="3936610">induction</span> <span m="3937050">hypothesis,</span> <span m="3937600">within</span> <span m="3937980">each</span> <span m="3938680">subnetwork,</span> <span m="3942290">we</span> <span m="3942390">can</span> <span m="3942530">solve</span> <span m="3942770">any</span> <span m="3943020">routing</span> <span m="3943360">problem</span> <span m="3943830">with</span> <span m="3944020">congestion</span> <span m="3944590">one</span> <span m="3945350">and</span> <span m="3945550">for</span> <span m="3945740">this</span> <span m="3946045">subnetwork,</span> <span m="3946640">the</span> <span m="3946740">same.</span> <span m="3947650">That's</span> <span m="3947820">our</span> <span m="3947960">induction</span> <span m="3948390">hypothesis.</span> </p>
<p><span m="3949990">So</span> <span m="3950150">how</span> <span m="3950300">do</span> <span m="3950390">we</span> <span m="3950500">go</span> <span m="3950740">ahead?</span> <span m="3951390">We</span> <span m="3951640">need</span> <span m="3951900">to</span> <span m="3954120">somehow</span> <span m="3954540">map</span> <span m="3954930">these</span> <span m="3955120">inputs</span> <span m="3955780">according</span> <span m="3956110">to</span> <span m="3956210">the</span> <span m="3956330">permutation</span> <span m="3957060">of</span> <span m="3957210">our</span> <span m="3957440">choice.</span> <span m="3958590">So</span> <span m="3958830">that</span> <span m="3959030">could</span> <span m="3959180">be</span> <span m="3959850">for</span> <span m="3960040">some</span> <span m="3960210">input</span> <span m="3960510">zero</span> <span m="3960780">goes</span> <span m="3961110">to</span> <span m="3961280">output</span> <span m="3961630">five</span> <span m="3962080">or</span> <span m="3962950">input</span> <span m="3963270">one</span> <span m="3963540">goes</span> <span m="3963850">to</span> <span m="3964140">output</span> <span m="3964850">two,</span> <span m="3965130">et</span> <span m="3965450">cetera.</span> <span m="3966580">So</span> <span m="3966920">somehow,</span> <span m="3967430">we</span> <span m="3967540">need</span> <span m="3967760">to</span> <span m="3967980">choose</span> <span m="3969080">where</span> <span m="3969420">we</span> <span m="3969590">are</span> <span m="3969660">going</span> <span m="3970000">to</span> <span m="3971000">map</span> <span m="3972140">this</span> <span m="3972340">particular</span> <span m="3972740">input</span> <span m="3973190">to.</span> <span m="3974120">So</span> <span m="3976150">packet</span> <span m="3976380">zero</span> <span m="3976910">that</span> <span m="3977100">comes</span> <span m="3977350">from</span> <span m="3977470">this</span> <span m="3977630">input</span> <span m="3978530">should</span> <span m="3978770">either</span> <span m="3979020">go</span> <span m="3979200">to</span> <span m="3979380">the</span> <span m="3979530">red</span> <span m="3980100">network</span> <span m="3981110">or it</span> <span m="3981290">should</span> <span m="3981490">go</span> <span m="3981630">to</span> <span m="3981760">the</span> <span m="3981850">blue</span> <span m="3982100">network.</span> <span m="3984100">And</span> <span m="3984980">for</span> <span m="3985250">each</span> <span m="3985570">of</span> <span m="3985680">these</span> <span m="3985880">inputs,</span> <span m="3986330">we</span> <span m="3986430">can</span> <span m="3986440">make</span> <span m="3986670">such</span> <span m="3986880">a</span> <span m="3986950">choice.</span> <span m="3988040">But</span> <span m="3988160">we</span> <span m="3988280">have</span> <span m="3988370">to</span> <span m="3988460">be</span> <span m="3988540">very</span> <span m="3988770">smart</span> <span m="3989180">about</span> <span m="3989490">it</span> <span m="3989540">because</span> <span m="3989810">we</span> <span m="3989880">need</span> <span m="3990070">to</span> <span m="3990200">avoid</span> <span m="3991060">any</span> <span m="3991300">congestion.</span> </p>
<p><span m="3993040">So</span> <span m="3993290">the</span> <span m="3993420">intuition</span> <span m="3993990">is</span> <span m="3994170">that</span> <span m="3994360">we're</span> <span m="3994640">going</span> <span m="3994970">to</span> <span m="3996310">set</span> <span m="3996550">up</span> <span m="3996690">a</span> <span m="3996770">constraint</span> <span m="3999280">graph,</span> <span m="4000190">a</span> <span m="4000270">graph</span> <span m="4001540">that</span> <span m="4003230">represents</span> <span m="4003990">all</span> <span m="4004170">the</span> <span m="4004260">constraints</span> <span m="4005240">that</span> <span m="4005380">we</span> <span m="4005470">need</span> <span m="4005630">to</span> <span m="4005720">satisfy</span> <span m="4006680">in</span> <span m="4006880">order</span> <span m="4007060">to</span> <span m="4007240">achieve</span> <span m="4007680">congestion</span> <span m="4008200">of</span> <span m="4008320">one.</span> <span m="4009990">So</span> <span m="4010220">let's</span> <span m="4010620">do</span> <span m="4010820">an</span> <span m="4010900">example</span> <span m="4012150">so</span> <span m="4012310">that</span> <span m="4012600">we</span> <span m="4012720">can</span> <span m="4013500">figure</span> <span m="4013840">out</span> <span m="4014010">what's</span> <span m="4014230">going</span> <span m="4014550">on.</span> <span m="4014910">Actually,</span> <span m="4015215">let</span> <span m="4015520">me</span> <span m="4015750">put</span> <span m="4015940">it</span> <span m="4016090">over</span> <span m="4016400">here.</span> <span m="4019540">So</span> <span m="4019770">just</span> <span m="4019980">take</span> <span m="4020170">an</span> <span m="4020230">example</span> <span m="4020670">permutation</span> <span m="4021970">and</span> <span m="4022090">we'll</span> <span m="4022200">go</span> <span m="4022400">through</span> <span m="4023130">this</span> <span m="4023310">example</span> <span m="4023970">and</span> <span m="4024090">then</span> <span m="4024220">see</span> <span m="4024510">how</span> <span m="4025320">the</span> <span m="4025490">proof</span> <span m="4025800">works.</span> </p>
<p><span m="4029240">So</span> <span m="4029490">let's</span> <span m="4029990">as</span> <span m="4030180">an</span> <span m="4030260">example</span> <span m="4030810">have</span> <span m="4031660">pi</span> <span m="4032060">of</span> <span m="4032420">zero</span> <span m="4032580">maps</span> <span m="4032880">to</span> <span m="4033020">one,</span> <span m="4034511">pi</span> <span m="4035008">of one</span> <span m="4036002">maps</span> <span m="4036500">to</span> <span m="4036660">five,</span> <span m="4038640">pi</span> <span m="4039135">of two</span> <span m="4039630">goes</span> <span m="4040130">to</span> <span m="4040270">four,</span> <span m="4042710">input</span> <span m="4043130">three</span> <span m="4043380">goes</span> <span m="4043660">to</span> <span m="4043780">seven,</span> <span m="4047000">four</span> <span m="4047415">maps</span> <span m="4047830">to</span> <span m="4047940">three,</span> <span m="4050826">five</span> <span m="4051310">to</span> <span m="4051600">six,</span> <span m="4053810">six</span> <span m="4054280">to</span> <span m="4054550">zero,</span> <span m="4056540">and</span> <span m="4056780">seven</span> <span m="4057170">to</span> <span m="4057360">two.</span> <span m="4057710">So</span> <span m="4057840">this is</span> <span m="4058020">just</span> <span m="4058220">an</span> <span m="4058320">arbitrary</span> <span m="4058810">permutation.</span> <span m="4064430">So</span> <span m="4065120">what</span> <span m="4065300">do</span> <span m="4065410">we</span> <span m="4065540">see?</span> <span m="4066060">We</span> <span m="4066250">want</span> <span m="4067480">to</span> <span m="4067650">make</span> <span m="4067850">sure</span> <span m="4068230">that,</span> <span m="4068440">for</span> <span m="4068530">example,</span> <span m="4068940">this</span> <span m="4069260">switch</span> <span m="4069990">is</span> <span m="4070360">only</span> <span m="4070820">seeing</span> <span m="4071230">one</span> <span m="4071560">packet.</span> <span m="4072460">So</span> <span m="4072700">it</span> <span m="4072800">cannot</span> <span m="4073220">see a</span> <span m="4073530">packet</span> <span m="4075090">both</span> <span m="4075560">coming</span> <span m="4075790">from</span> <span m="4075960">input</span> <span m="4076250">zero</span> <span m="4076990">as</span> <span m="4077180">well</span> <span m="4077490">as</span> <span m="4077640">from</span> <span m="4077820">input</span> <span m="4078210">four.</span> <span m="4079220">I</span> <span m="4079300">cannot</span> <span m="4079640">see</span> <span m="4079880">that.</span> <span m="4080670">I</span> <span m="4080950">do</span> <span m="4081050">not</span> <span m="4081210">want</span> <span m="4081400">that</span> <span m="4081790">to</span> <span m="4081920">happen.</span> </p>
<p><span m="4082540">Similarly,</span> <span m="4082980">for</span> <span m="4083090">this</span> <span m="4083340">one,</span> <span m="4083910">I</span> <span m="4084000">do not</span> <span m="4084260">want</span> <span m="4084390">to</span> <span m="4084470">see</span> <span m="4084650">a</span> <span m="4084800">packet</span> <span m="4085060">coming</span> <span m="4085330">from</span> <span m="4085590">one</span> <span m="4086450">or</span> <span m="4086820">one</span> <span m="4087200">from</span> <span m="4087660">five.</span> <span m="4089550">So</span> <span m="4089790">let</span> <span m="4089950">me</span> <span m="4090070">define</span> <span m="4090460">a</span> <span m="4090530">constraint</span> <span m="4091140">graph</span> <span m="4091680">that</span> <span m="4091770">sort</span> <span m="4092000">of</span> <span m="4092110">represents</span> <span m="4092780">this.</span> <span m="4094500">So</span> <span m="4097160">the</span> <span m="4097279">constraint</span> <span m="4097870">graph</span> <span m="4098490">that</span> <span m="4098680">we are</span> <span m="4098920">interested</span> <span m="4099640">in</span> <span m="4102029">is</span> <span m="4102189">defined</span> <span m="4102609">as</span> <span m="4102750">follows.</span> <span m="4107479">If</span> <span m="4107790">two</span> <span m="4107899">packets</span> <span m="4116050">must</span> <span m="4116319">pass</span> <span m="4121630">through</span> <span m="4121859">different</span> <span m="4122189">networks,</span> <span m="4124359">subnetworks--</span> <span m="4128560">so</span> <span m="4128720">in</span> <span m="4128819">our</span> <span m="4129029">case,</span> <span m="4129300">the</span> <span m="4129370">red</span> <span m="4129560">and</span> <span m="4129740">blue</span> <span m="4129890">subnetwork--</span> <span m="4136569">then we'll</span> <span m="4136859">actually</span> <span m="4138210">have</span> <span m="4138450">an</span> <span m="4138569">edge</span> <span m="4138890">between</span> <span m="4139270">those</span> <span m="4139560">two.</span> <span m="4140600">So</span> <span m="4142810">then,</span> <span m="4144830">there</span> <span m="4145100">is</span> <span m="4146670">an</span> <span m="4146950">edge</span> <span m="4149060">between</span> <span m="4149450">them.</span> <span m="4150470">So</span> <span m="4150580">for</span> <span m="4150700">this</span> <span m="4150890">example,</span> <span m="4151740">we're</span> <span m="4151840">going</span> <span m="4151990">to</span> <span m="4152100">set</span> <span m="4152330">up</span> <span m="4152470">this</span> <span m="4152649">constraint</span> <span m="4153149">graph.</span> </p>
<p><span m="4173600">So</span> <span m="4174160">I</span> <span m="4174200">was</span> <span m="4174359">just</span> <span m="4174600">talking</span> <span m="4174970">about</span> <span m="4175260">this</span> <span m="4175420">particular</span> <span m="4175859">switch.</span> <span m="4177029">It</span> <span m="4177229">cannot</span> <span m="4177569">see</span> <span m="4177830">one</span> <span m="4178040">coming</span> <span m="4178290">from</span> <span m="4178470">four</span> <span m="4178950">and</span> <span m="4179390">a</span> <span m="4179550">packet</span> <span m="4179700">from</span> <span m="4180069">zero.</span> <span m="4181020">So</span> <span m="4182479">what</span> <span m="4182649">he</span> <span m="4182740">have,</span> <span m="4183130">we</span> <span m="4183359">have</span> <span m="4183510">an</span> <span m="4183680">edge</span> <span m="4184380">between</span> <span m="4184720">zero</span> <span m="4185500">and</span> <span m="4185720">four.</span> <span m="4187140">In</span> <span m="4187270">the</span> <span m="4187359">same</span> <span m="4188140">way,</span> <span m="4188569">we</span> <span m="4188710">have</span> <span m="4188950">an</span> <span m="4189090">edge</span> <span m="4190020">from</span> <span m="4191279">one</span> <span m="4191970">to</span> <span m="4192109">five.</span> <span m="4192539">Why?</span> <span m="4193390">Because</span> <span m="4193729">a</span> <span m="4193800">packet</span> <span m="4195350">that</span> <span m="4195530">comes from</span> <span m="4196010">input</span> <span m="4196200">one</span> <span m="4196940">and</span> <span m="4197180">a packet</span> <span m="4197510">that comes</span> <span m="4199000">from</span> <span m="4199160">input</span> <span m="4199510">five</span> <span m="4199910">cannot</span> <span m="4200600">both</span> <span m="4201010">be</span> <span m="4201190">routed</span> <span m="4201900">through</span> <span m="4202150">the</span> <span m="4202270">switch</span> <span m="4202820">because</span> <span m="4203120">then</span> <span m="4203220">the</span> <span m="4203310">switch</span> <span m="4203680">would</span> <span m="4203840">see</span> <span m="4204100">two</span> <span m="4204380">packets</span> <span m="4205370">and</span> <span m="4205500">then the</span> <span m="4205770">congestion</span> <span m="4206370">would</span> <span m="4206580">not be</span> <span m="4206810">one,</span> <span m="4207010">but</span> <span m="4207190">two,</span> <span m="4207440">right?</span> </p>
<p><span m="4208220">So</span> <span m="4209200">one</span> <span m="4209610">and</span> <span m="4209850">five</span> <span m="4210510">also</span> <span m="4210850">have</span> <span m="4211000">an edge in</span> <span m="4211430">between.</span> <span m="4215220">And</span> <span m="4215500">in</span> <span m="4215630">the</span> <span m="4215710">same</span> <span m="4216060">way,</span> <span m="4216470">we</span> <span m="4216560">have</span> <span m="4216950">two</span> <span m="4217370">and</span> <span m="4217550">six</span> <span m="4218510">and</span> <span m="4218740">seven</span> <span m="4219790">and</span> <span m="4220210">three.</span> <span m="4220760">So</span> <span m="4220970">two</span> <span m="4221200">and</span> <span m="4221310">six</span> <span m="4225120">is</span> <span m="4225280">this</span> <span m="4225470">constraint,</span> <span m="4226260">like</span> <span m="4226520">two</span> <span m="4227300">and</span> <span m="4227560">six</span> <span m="4228520">over</span> <span m="4228830">here.</span> <span m="4229680">And</span> <span m="4229890">three</span> <span m="4230720">and</span> <span m="4230930">seven</span> <span m="4231370">is</span> <span m="4231480">the</span> <span m="4231690">other</span> <span m="4231800">constraint.</span> <span m="4232250">So</span> <span m="4232640">if</span> <span m="4232860">I</span> <span m="4232970">have</span> <span m="4233200">those</span> <span m="4233480">constraints</span> <span m="4234110">in</span> <span m="4234220">place,</span> <span m="4236110">well</span> <span m="4236290">then,</span> <span m="4236470">I</span> <span m="4236550">know</span> <span m="4237070">that</span> <span m="4238450">the</span> <span m="4238570">routing</span> <span m="4239260">that</span> <span m="4239440">goes</span> <span m="4239810">from</span> <span m="4240420">level</span> <span m="4240730">zero</span> <span m="4241540">to</span> <span m="4241680">level</span> <span m="4242080">one</span> <span m="4242970">will</span> <span m="4243130">not</span> <span m="4243330">violate</span> <span m="4244040">my</span> <span m="4244500">congestion</span> <span m="4244875">of</span> <span m="4245250">one.</span> <span m="4246580">So</span> <span m="4246730">that's</span> <span m="4247030">great.</span> </p>
<p><span m="4247910">Then,</span> <span m="4248090">I</span> <span m="4248190">hope</span> <span m="4248400">to</span> <span m="4248530">be</span> <span m="4248670">able</span> <span m="4248830">to</span> <span m="4248950">use the</span> <span m="4249190">induction</span> <span m="4249770">hypothesis</span> <span m="4251110">and</span> <span m="4251280">I</span> <span m="4251340">get</span> <span m="4251540">a</span> <span m="4251610">proper</span> <span m="4251940">routing</span> <span m="4252440">within</span> <span m="4252850">the</span> <span m="4252970">red</span> <span m="4253230">subnetwork</span> <span m="4254160">and</span> <span m="4254320">one</span> <span m="4254600">within</span> <span m="4255120">the</span> <span m="4255220">blue</span> <span m="4255450">network.</span> <span m="4257110">And</span> <span m="4257230">then,</span> <span m="4257390">I</span> <span m="4257450">need</span> <span m="4257660">to</span> <span m="4257760">map</span> <span m="4258070">all</span> <span m="4258330">these</span> <span m="4258803">to</span> <span m="4259276">these</span> <span m="4259750">outputs.</span> <span m="4260770">So</span> <span m="4261130">I</span> <span m="4261330">also</span> <span m="4261670">have</span> <span m="4261840">constraints</span> <span m="4262350">on</span> <span m="4262420">these</span> <span m="4262660">outputs</span> <span m="4263210">because,</span> <span m="4265740">well,</span> <span m="4267040">For</span> <span m="4267150">example,</span> <span m="4267650">take</span> <span m="4267900">this</span> <span m="4268080">particular</span> <span m="4268510">switch.</span> <span m="4269490">It</span> <span m="4269650">should</span> <span m="4269880">not</span> <span m="4270130">see</span> <span m="4271200">a</span> <span m="4271320">packet</span> <span m="4271830">coming</span> <span m="4272130">from</span> <span m="4272380">this</span> <span m="4272600">particular</span> <span m="4273080">one</span> <span m="4273460">and</span> <span m="4273670">one</span> <span m="4273850">from</span> <span m="4274040">this</span> <span m="4274290">one.</span> <span m="4275220">So</span> <span m="4275330">how</span> <span m="4275490">do</span> <span m="4275610">I</span> <span m="4275690">code</span> <span m="4275970">that</span> <span m="4276150">up?</span> </p>
<p><span m="4283950">So</span> <span m="4284130">let</span> <span m="4284370">me</span> <span m="4284950">first</span> <span m="4285330">write</span> <span m="4285540">out</span> <span m="4285810">what</span> <span m="4285950">we</span> <span m="4286070">did</span> <span m="4286250">here</span> <span m="4287070">and</span> <span m="4287170">then</span> <span m="4287290">we'll</span> <span m="4287380">do</span> <span m="4287500">the</span> <span m="4287610">same</span> <span m="4287960">for</span> <span m="4288800">the</span> <span m="4288930">last</span> <span m="4289490">level</span> <span m="4289780">over</span> <span m="4290120">there.</span> <span m="4292090">So--</span> <span m="4293300">oh no,</span> <span m="4293575">that's</span> <span m="4293850">not</span> <span m="4294040">really</span> <span m="4294440">necessary.</span> <span m="4301770">So</span> <span m="4301940">at</span> <span m="4302070">the</span> <span m="4302180">output</span> <span m="4302530">side</span> <span m="4302840">over</span> <span m="4303090">here,</span> <span m="4303590">we</span> <span m="4303720">have</span> <span m="4303950">similar</span> <span m="4304370">constraints</span> <span m="4305020">as</span> <span m="4305170">we</span> <span m="4305260">did</span> <span m="4305450">over</span> <span m="4305700">here.</span> <span m="4306710">And</span> <span m="4306810">in</span> <span m="4306900">this</span> <span m="4307090">particular</span> <span m="4307440">example,</span> <span m="4308740">just</span> <span m="4308850">as</span> <span m="4308960">an</span> <span m="4309220">example,</span> <span m="4311800">suppose</span> <span m="4312280">we</span> <span m="4312430">look</span> <span m="4312640">at</span> <span m="4312720">the</span> <span m="4312820">packet</span> <span m="4315370">that</span> <span m="4315630">is</span> <span m="4316330">destined</span> <span m="4319720">for</span> <span m="4321440">output</span> <span m="4322170">zero.</span> </p>
<p><span m="4324910">Well,</span> <span m="4325110">what</span> <span m="4325250">is</span> <span m="4325420">this</span> <span m="4325640">packet?</span> <span m="4325910">Well,</span> <span m="4326220">I</span> <span m="4326300">know</span> <span m="4326610">that's</span> <span m="4327110">pi</span> <span m="4328120">of</span> <span m="4328330">6</span> <span m="4328860">is</span> <span m="4329040">equal</span> <span m="4329440">to</span> <span m="4329730">0,</span> <span m="4330620">according</span> <span m="4331020">to</span> <span m="4331170">my</span> <span m="4331330">example.</span> <span m="4332320">So</span> <span m="4333050">packet</span> <span m="4333380">six</span> <span m="4334550">is</span> <span m="4334820">destined</span> <span m="4335280">for</span> <span m="4335420">this</span> <span m="4335610">particular</span> <span m="4336100">output</span> <span m="4336530">zero</span> <span m="4337980">over</span> <span m="4338270">here</span> <span m="4338900">and</span> <span m="4340190">goes</span> <span m="4340470">through</span> <span m="4340660">this</span> <span m="4340880">particular</span> <span m="4341260">switch.</span> <span m="4343540">So</span> <span m="4343630">this</span> <span m="4343960">packet</span> <span m="4347980">and</span> <span m="4348230">also</span> <span m="4353401">the</span> <span m="4353860">packet</span> <span m="4359890">for</span> <span m="4360336">output</span> <span m="4361540">four,</span> <span m="4363450">which</span> <span m="4363740">is</span> <span m="4364290">if</span> <span m="4364470">you</span> <span m="4364600">look</span> <span m="4364790">at</span> <span m="4364920">the</span> <span m="4365010">mapping,</span> <span m="4365740">pi</span> <span m="4366000">of</span> <span m="4366160">2</span> <span m="4366820">is</span> <span m="4367060">equal</span> <span m="4367350">to</span> <span m="4367480">4.</span> <span m="4368420">So</span> <span m="4368580">that's</span> <span m="4368910">packet</span> <span m="4369170">number</span> <span m="4369500">two.</span> <span m="4371360">Well,</span> <span m="4372340">both</span> <span m="4372730">of</span> <span m="4372850">these</span> <span m="4373130">packets</span> <span m="4374350">cannot</span> <span m="4374850">pass</span> <span m="4375370">through</span> <span m="4375530">the</span> <span m="4375650">same</span> <span m="4375940">subnetwork.</span> </p>
<p><span m="4386990">So</span> <span m="4387180">why</span> <span m="4387400">is</span> <span m="4387570">this?</span> <span m="4388320">So</span> <span m="4388530">let's</span> <span m="4388980">look</span> <span m="4389230">at</span> <span m="4389270">this</span> <span m="4389410">particular</span> <span m="4389780">example.</span> <span m="4391020">So</span> <span m="4392230">output</span> <span m="4392700">zero,</span> <span m="4394880">well,</span> <span m="4395260">comes</span> <span m="4395640">from</span> <span m="4396940">packet</span> <span m="4397610">six,</span> <span m="4398680">somewhere</span> <span m="4399110">over</span> <span m="4399350">there.</span> <span m="4400315">Now</span> <span m="4400700">suppose</span> <span m="4401480">packet</span> <span m="4401780">six</span> <span m="4402530">was</span> <span m="4402740">routed</span> <span m="4403120">through</span> <span m="4403270">the</span> <span m="4403440">red</span> <span m="4403930">network</span> <span m="4405780">and</span> <span m="4406280">at</span> <span m="4406410">the</span> <span m="4406520">same</span> <span m="4406880">moment</span> <span m="4407840">also,</span> <span m="4409160">output</span> <span m="4409600">four--</span> <span m="4410000">the</span> <span m="4410150">packet</span> <span m="4410425">that is</span> <span m="4410700">destined</span> <span m="4410990">for output</span> <span m="4411360">four,</span> <span m="4411730">which</span> <span m="4411990">is packet</span> <span m="4412250">number</span> <span m="4412710">two--</span> <span m="4414030">suppose</span> <span m="4414540">packet</span> <span m="4414960">two</span> <span m="4415110">was</span> <span m="4415350">also</span> <span m="4416890">going</span> <span m="4417190">through</span> <span m="4417370">the</span> <span m="4417530">red</span> <span m="4417960">network.</span> </p>
<p><span m="4420010">Well,</span> <span m="4422430">then</span> <span m="4423320">I</span> <span m="4423480">notice</span> <span m="4424590">that</span> <span m="4425270">both</span> <span m="4425840">of</span> <span m="4426040">these</span> <span m="4426280">packets</span> <span m="4426720">must</span> <span m="4427280">arrive</span> <span m="4427690">at</span> <span m="4427820">this</span> <span m="4428010">particular</span> <span m="4428470">switch</span> <span m="4429830">in</span> <span m="4430040">order</span> <span m="4430760">for</span> <span m="4431010">one</span> <span m="4431260">to</span> <span m="4431380">be</span> <span m="4431540">routed</span> <span m="4432210">to</span> <span m="4432430">output</span> <span m="4432760">zero</span> <span m="4433880">and</span> <span m="4434240">the</span> <span m="4434370">other</span> <span m="4434420">one</span> <span m="4434640">to</span> <span m="4434740">be</span> <span m="4434910">routed</span> <span m="4435490">to</span> <span m="4435830">output</span> <span m="4436200">four.</span> <span m="4437890">So</span> <span m="4438080">in</span> <span m="4438160">order</span> <span m="4438390">to</span> <span m="4438560">avoid</span> <span m="4439110">congestion</span> <span m="4440130">in</span> <span m="4440260">this</span> <span m="4440450">particular</span> <span m="4441040">switch</span> <span m="4441470">over</span> <span m="4441760">here,</span> <span m="4442790">we</span> <span m="4442950">need</span> <span m="4443200">to</span> <span m="4443540">have</span> <span m="4443730">a</span> <span m="4443790">constraint.</span> <span m="4444890">The</span> <span m="4444980">constraint</span> <span m="4445460">says</span> <span m="4445800">that</span> <span m="4445880">the</span> <span m="4445970">packet</span> <span m="4446380">for</span> <span m="4448290">packets</span> <span m="4448700">two</span> <span m="4449630">and</span> <span m="4449870">six,</span> <span m="4451150">that</span> <span m="4451280">those</span> <span m="4451590">two</span> <span m="4451830">cannot</span> <span m="4452250">go</span> <span m="4452480">through</span> <span m="4452670">the</span> <span m="4452810">same</span> <span m="4453340">subnetwork.</span> <span m="4454840">So</span> <span m="4454960">essentially</span> <span m="4455480">have</span> <span m="4457090">another</span> <span m="4457450">edge</span> <span m="4457660">over</span> <span m="4458040">here--</span> <span m="4458200">we</span> <span m="4458300">already</span> <span m="4458630">had</span> <span m="4458830">the</span> <span m="4458910">constraint</span> <span m="4460930">but</span> <span m="4461080">it's</span> <span m="4462020">just</span> <span m="4462190">the</span> <span m="4462260">same</span> <span m="4462570">edge.</span> <span m="4463640">So</span> <span m="4463810">let's</span> <span m="4464060">look</span> <span m="4464270">at</span> <span m="4464500">the</span> <span m="4464880">other</span> <span m="4466550">constraints</span> <span m="4467290">that</span> <span m="4467350">we</span> <span m="4467450">have.</span> </p>
<p><span m="4473770">Well,</span> <span m="4474170">let's</span> <span m="4474750">look</span> <span m="4474980">at</span> <span m="4475050">a</span> <span m="4475110">different</span> <span m="4475420">example.</span> <span m="4475960">So</span> <span m="4476100">for</span> <span m="4476240">example,</span> <span m="4477860">if</span> <span m="4478080">I</span> <span m="4478160">look</span> <span m="4478660">at</span> <span m="4479860">this</span> <span m="4480130">switch,</span> <span m="4481110">well,</span> <span m="4483250">if</span> <span m="4484730">a</span> <span m="4484830">packet</span> <span m="4485340">goes</span> <span m="4485530">through</span> <span m="4485980">here</span> <span m="4486110">that needs</span> <span m="4486340">to</span> <span m="4486440">end</span> <span m="4486680">up</span> <span m="4487740">at</span> <span m="4487970">one</span> <span m="4488420">and</span> <span m="4488680">a</span> <span m="4488740">packet</span> <span m="4489110">that's</span> <span m="4489300">goes</span> <span m="4489990">to</span> <span m="4490150">five,</span> <span m="4491800">if</span> <span m="4492240">those</span> <span m="4492520">two</span> <span m="4492660">packets</span> <span m="4493530">are</span> <span m="4494090">routed</span> <span m="4494500">through</span> <span m="4494670">the</span> <span m="4494820">same</span> <span m="4495350">red</span> <span m="4495730">subnetwork,</span> <span m="4496600">they</span> <span m="4496730">have</span> <span m="4497060">to</span> <span m="4497170">end</span> <span m="4497370">up</span> <span m="4497550">here</span> <span m="4498050">in</span> <span m="4498230">order</span> <span m="4498410">to</span> <span m="4498530">go</span> <span m="4498760">to</span> <span m="4498910">both</span> <span m="4499290">here</span> <span m="4499990">and</span> <span m="4500190">to</span> <span m="4500320">there.</span> <span m="4500910">So</span> <span m="4501080">we have</span> <span m="4501210">congestion</span> <span m="4501700">of</span> <span m="4501820">two.</span> <span m="4502520">So</span> <span m="4503110">what</span> <span m="4503290">are</span> <span m="4503380">those</span> <span m="4503680">packets?</span> <span m="4504780">Well,</span> <span m="4506120">what</span> <span m="4506300">does</span> <span m="4506890">pi</span> <span m="4507175">map</span> <span m="4507460">to</span> <span m="4508270">to</span> <span m="4508510">one</span> <span m="4508750">and</span> <span m="4508900">five?</span> </p>
<p><span m="4509810">Let's</span> <span m="4510050">look</span> <span m="4510220">over</span> <span m="4510480">here.</span> <span m="4511280">We</span> <span m="4511380">see</span> <span m="4511580">that</span> <span m="4511830">pi</span> <span m="4514030">0</span> <span m="4514770">is</span> <span m="4514940">equal</span> <span m="4515240">to</span> <span m="4515390">1</span> <span m="4516070">and</span> <span m="4516290">pi</span> <span m="4517740">1</span> <span m="4518160">is</span> <span m="4518330">equal</span> <span m="4518540">to</span> <span m="4518670">5.</span> <span m="4519250">So</span> <span m="4519750">packets</span> <span m="4520100">zero</span> <span m="4520560">and</span> <span m="4520700">one</span> <span m="4521610">are</span> <span m="4521780">actually</span> <span m="4522100">mapped</span> <span m="4522380">to</span> <span m="4522550">output</span> <span m="4523810">one</span> <span m="4524260">and</span> <span m="4524490">five</span> <span m="4524940">and</span> <span m="4525070">they</span> <span m="4525140">should</span> <span m="4525340">not</span> <span m="4525550">go</span> <span m="4525710">both</span> <span m="4526320">through</span> <span m="4526490">the</span> <span m="4526630">same</span> <span m="4526960">subnetwork.</span> <span m="4529490">So</span> <span m="4529510">we</span> <span m="4529600">have</span> <span m="4529730">another</span> <span m="4530060">edge</span> <span m="4530550">over</span> <span m="4530860">here.</span> <span m="4532132">And</span> <span m="4532540">now,</span> <span m="4532760">we</span> <span m="4532870">can</span> <span m="4533040">continue</span> <span m="4533530">this</span> <span m="4534990">and</span> <span m="4535280">we</span> <span m="4535400">have</span> <span m="4535620">five</span> <span m="4535960">and</span> <span m="4536080">seven.</span> <span m="4536740">So</span> <span m="4537010">just</span> <span m="4537210">have</span> <span m="4537320">a</span> <span m="4537390">look</span> <span m="4537580">over</span> <span m="4537800">there.</span> <span m="4537990">See,</span> <span m="4538150">five</span> <span m="4538490">and seven,</span> <span m="4539450">they</span> <span m="4539570">map</span> <span m="4539890">to</span> <span m="4540730">the</span> <span m="4540960">outputs</span> <span m="4541510">two</span> <span m="4541720">and</span> <span m="4541830">six.</span> <span m="4542320">Again,</span> <span m="4542640">we</span> <span m="4542740">have</span> <span m="4543300">two</span> <span m="4544030">and</span> <span m="4544230">six.</span> <span m="4544650">If</span> <span m="4544790">they</span> <span m="4544970">are</span> <span m="4545080">both</span> <span m="4545820">mapped</span> <span m="4546160">to</span> <span m="4546260">the</span> <span m="4546360">same</span> <span m="4546680">network,</span> <span m="4547400">this</span> <span m="4547650">one,</span> <span m="4547900">for</span> <span m="4548070">example,</span> <span m="4550140">then</span> <span m="4550340">I</span> <span m="4550400">will</span> <span m="4550550">have a</span> <span m="4550800">problem.</span> <span m="4553870">So</span> <span m="4555420">the</span> <span m="4556030">other</span> <span m="4556560">edge</span> <span m="4556950">is</span> <span m="4557230">over</span> <span m="4557560">here.</span> </p>
<p><span m="4560580">So</span> <span m="4560800">what</span> <span m="4560950">did</span> <span m="4561120">we</span> <span m="4561230">do</span> <span m="4561500">here?</span> <span m="4562070">We</span> <span m="4562160">started</span> <span m="4562570">to</span> <span m="4562800">write</span> <span m="4563040">out</span> <span m="4563200">the</span> <span m="4563280">constraints</span> <span m="4564200">on</span> <span m="4564390">this</span> <span m="4564765">side</span> <span m="4565810">and we</span> <span m="4566160">wrote</span> <span m="4566610">out</span> <span m="4566780">the</span> <span m="4566850">constraints</span> <span m="4567410">on</span> <span m="4567590">this</span> <span m="4567840">side.</span> <span m="4570580">So</span> <span m="4570870">I</span> <span m="4571020">only</span> <span m="4571290">looked</span> <span m="4571540">at</span> <span m="4571630">the</span> <span m="4571730">red</span> <span m="4572060">subnetwork.</span> <span m="4572770">That's</span> <span m="4572930">what</span> <span m="4573090">I</span> <span m="4573140">realize</span> <span m="4573670">now.</span> <span m="4573920">I</span> <span m="4573970">could</span> <span m="4574150">also</span> <span m="4574350">have</span> <span m="4574480">looked</span> <span m="4574690">at</span> <span m="4574790">the</span> <span m="4574870">blue</span> <span m="4575170">network.</span> <span m="4576260">So</span> <span m="4576440">let's</span> <span m="4576640">do</span> <span m="4576750">that</span> <span m="4576950">also</span> <span m="4577500">just</span> <span m="4577720">to</span> <span m="4577820">make</span> <span m="4578110">the</span> <span m="4578200">picture</span> <span m="4578540">complete.</span> <span m="4579390">So</span> <span m="4579520">for</span> <span m="4579680">example,</span> <span m="4580280">let's</span> <span m="4580420">look</span> <span m="4580650">at</span> <span m="4581760">this</span> <span m="4581910">particular</span> <span m="4582330">example.</span> <span m="4584010">The</span> <span m="4584140">packet</span> <span m="4584510">six</span> <span m="4584870">and</span> <span m="4585010">two</span> <span m="4585790">should</span> <span m="4586030">not</span> <span m="4586280">both</span> <span m="4586770">be</span> <span m="4586940">routed</span> <span m="4587710">through</span> <span m="4588250">the</span> <span m="4588340">blue</span> <span m="4588680">network</span> <span m="4589560">because</span> <span m="4591100">then</span> <span m="4591300">they</span> <span m="4591430">would</span> <span m="4591670">both</span> <span m="4591970">have</span> <span m="4592150">to</span> <span m="4592250">go</span> <span m="4592430">through</span> <span m="4592600">this</span> <span m="4592860">switch,</span> <span m="4594000">one</span> <span m="4594480">going</span> <span m="4594780">up</span> <span m="4594980">to</span> <span m="4595190">output</span> <span m="4595480">zero</span> <span m="4595990">and</span> <span m="4596130">one</span> <span m="4596310">going</span> <span m="4596520">to</span> <span m="4596800">the</span> <span m="4597800">right</span> <span m="4598150">to</span> <span m="4598390">output</span> <span m="4598720">four.</span> </p>
<p><span m="4600300">So</span> <span m="4600890">in order</span> <span m="4600960">to</span> <span m="4601100">avoid</span> <span m="4601680">congestion</span> <span m="4602760">at</span> <span m="4602910">all</span> <span m="4603090">costs,</span> <span m="4603670">we</span> <span m="4603780">have</span> <span m="4603940">this</span> <span m="4604120">constraint</span> <span m="4604650">graph.</span> <span m="4605950">So</span> <span m="4606120">now,</span> <span m="4606310">we</span> <span m="4606440">come</span> <span m="4606630">to</span> <span m="4606790">the</span> <span m="4606920">key</span> <span m="4607200">insight.</span> <span m="4608500">And</span> <span m="4608630">the</span> <span m="4608710">key</span> <span m="4608940">insight</span> <span m="4609420">is</span> <span m="4609880">to</span> <span m="4609990">use</span> <span m="4610230">a</span> <span m="4610350">two-coloring</span> <span m="4611050">of</span> <span m="4612060">this</span> <span m="4612570">graph.</span> <span m="4617060">So</span> <span m="4617440">the</span> <span m="4617560">key</span> <span m="4617860">insight</span> <span m="4620360">is</span> <span m="4621080">a</span> <span m="4621950">two-coloring</span> <span m="4627110">of</span> <span m="4627840">the</span> <span m="4627980">constraint</span> <span m="4628300">graph,</span> <span m="4632680">which will</span> <span m="4632950">lead</span> <span m="4633490">to</span> <span m="4634240">a</span> <span m="4635020">best</span> <span m="4635440">solution</span> <span m="4636310">for</span> <span m="4636670">the routing</span> <span m="4637130">problem.</span> <span m="4639130">So</span> <span m="4640410">let's</span> <span m="4641020">do</span> <span m="4641280">this.</span> </p>
<p><span m="4642270">So</span> <span m="4642670">we</span> <span m="4643070">will</span> <span m="4643610">color</span> <span m="4643980">this</span> <span m="4644210">one</span> <span m="4644440">blue.</span> <span m="4645220">As</span> <span m="4645350">you</span> <span m="4645460">can</span> <span m="4645600">see,</span> <span m="4645770">this</span> <span m="4645960">is</span> <span m="4646110">an even</span> <span m="4646440">cycle,</span> <span m="4648210">blue,</span> <span m="4648960">red,</span> <span m="4650690">blue,</span> <span m="4651940">red,</span> <span m="4653650">and</span> <span m="4655020">blue</span> <span m="4655690">and</span> <span m="4655920">red.</span> <span m="4657030">We will</span> <span m="4657160">make</span> <span m="4657370">this</span> <span m="4657580">one</span> <span m="4657750">blue</span> <span m="4658300">and</span> <span m="4658440">this</span> <span m="4658650">one</span> <span m="4658860">red.</span> <span m="4660920">Well,</span> <span m="4661640">it</span> <span m="4661800">turns</span> <span m="4662170">out</span> <span m="4662600">that</span> <span m="4662880">we</span> <span m="4662980">can</span> <span m="4663120">now</span> <span m="4663600">start</span> <span m="4664270">our</span> <span m="4664510">routing</span> <span m="4664890">process.</span> <span m="4667170">So</span> <span m="4667400">for</span> <span m="4667610">example,</span> <span m="4669210">actually,</span> <span m="4670240">I</span> <span m="4670350">will</span> <span m="4670490">draw</span> <span m="4671560">a</span> <span m="4671640">new</span> <span m="4671890">graph</span> <span m="4672660">to</span> <span m="4672970">make</span> <span m="4673190">that</span> <span m="4673360">really</span> <span m="4673680">clear.</span> <span m="4683640">So</span> <span m="4683870">I</span> <span m="4683970">have</span> <span m="4684230">my</span> <span m="4684390">blue</span> <span m="4685820">and</span> <span m="4685990">my</span> <span m="4686180">red</span> <span m="4686910">chalk</span> <span m="4687270">over</span> <span m="4687550">here</span> <span m="4687740">to</span> <span m="4687960">demonstrate</span> <span m="4688720">what</span> <span m="4688820">I</span> <span m="4688930">mean.</span> </p>
<p><span m="4692520">So</span> <span m="4692770">what</span> <span m="4693090">do</span> <span m="4693250">I</span> <span m="4693320">do?</span> <span m="4693590">I</span> <span m="4693690">have</span> <span m="4693850">zero,</span> <span m="4694300">one,</span> <span m="4694750">two,</span> <span m="4695200">three,</span> <span m="4695650">four,</span> <span m="4696100">five,</span> <span m="4696550">six,</span> <span m="4697450">and</span> <span m="4697670">seven.</span> <span m="4699970">I</span> <span m="4700210">have</span> <span m="4700550">the</span> <span m="4700670">switches</span> <span m="4701100">that</span> <span m="4701410">correspond</span> <span m="4702030">to</span> <span m="4702150">those.</span> <span m="4706390">Well,</span> <span m="4706760">if</span> <span m="4707480">it's</span> <span m="4707910">colored</span> <span m="4709670">red--</span> <span m="4711030">so</span> <span m="4711630">zero</span> <span m="4711980">over</span> <span m="4712300">here</span> <span m="4712490">is</span> <span m="4712670">colored</span> <span m="4712990">red--</span> <span m="4713660">I will</span> <span m="4713870">direct it</span> <span m="4714310">to</span> <span m="4714530">the</span> <span m="4714660">red</span> <span m="4714950">subnetwork.</span> <span m="4715535">So</span> <span m="4715850">where is</span> <span m="4716300">this</span> <span m="4716440">red</span> <span m="4716690">subnetwork?</span> <span m="4718810">It's</span> <span m="4718960">really</span> <span m="4719240">contained</span> <span m="4721150">over</span> <span m="4721590">here</span> <span m="4725850">and</span> <span m="4726040">the</span> <span m="4726110">blue</span> <span m="4726500">one--</span> <span m="4728800">so</span> <span m="4728980">this</span> <span m="4729190">is</span> <span m="4729330">the</span> <span m="4729440">red</span> <span m="4729720">one</span> <span m="4732040">and</span> <span m="4732190">the</span> <span m="4732260">blue</span> <span m="4732500">one</span> <span m="4732880">is right</span> <span m="4733366">here.</span> <span m="4742120">And</span> <span m="4742300">over</span> <span m="4742530">here,</span> <span m="4742690">we</span> <span m="4742790">have</span> <span m="4743010">the</span> <span m="4743150">outputs</span> <span m="4747980">ranging</span> <span m="4748330">from</span> <span m="4748470">zero,</span> <span m="4748970">one,</span> <span m="4749320">two,</span> <span m="4750210">all</span> <span m="4750430">the</span> <span m="4750540">way</span> <span m="4752290">to</span> <span m="4752470">seven.</span> </p>
<p><span m="4754920">So</span> <span m="4755270">input</span> <span m="4755620">zero</span> <span m="4756120">is</span> <span m="4756310">colored</span> <span m="4756600">red.</span> <span m="4757530">We</span> <span m="4757640">go</span> <span m="4757790">straight</span> <span m="4758120">ahead.</span> <span m="4758380">We</span> <span m="4758500">want</span> <span m="4758630">to</span> <span m="4758730">go</span> <span m="4758900">to the</span> <span m="4759030">red</span> <span m="4759360">network.</span> <span m="4760550">Input</span> <span m="4760920">one</span> <span m="4761620">is</span> <span m="4761880">colored</span> <span m="4762300">blue.</span> <span m="4763350">It</span> <span m="4763540">goes,</span> <span m="4763930">therefore,</span> <span m="4765600">to</span> <span m="4765820">the</span> <span m="4765920">blue</span> <span m="4766140">network.</span> <span m="4767000">So</span> <span m="4767160">this</span> <span m="4767280">is  the</span> <span m="4767500">only</span> <span m="4767830">way</span> <span m="4768040">how to</span> <span m="4768090">do</span> <span m="4768270">it.</span> </p>
<p><span m="4769270">Input</span> <span m="4769630">two</span> <span m="4770010">is</span> <span m="4770210">colored</span> <span m="4770410">red.</span> <span m="4771640">Go</span> <span m="4771760">straight</span> <span m="4772140">ahead.</span> <span m="4773070">Input</span> <span m="4773390">three</span> <span m="4773800">is</span> <span m="4773970">also</span> <span m="4774250">colored</span> <span m="4774480">red.</span> <span m="4775330">Go</span> <span m="4775480">straight</span> <span m="4775890">ahead.</span> <span m="4776670">Input</span> <span m="4777060">five--</span> <span m="4778770">oh,</span> <span m="4778830">input</span> <span m="4779180">four</span> <span m="4780690">is</span> <span m="4780910">colored</span> <span m="4781170">blue--</span> <span m="4782180">goes</span> <span m="4782430">to</span> <span m="4782520">the</span> <span m="4782610">blue</span> <span m="4782810">network.</span> <span m="4783280">Input</span> <span m="4783980">five</span> <span m="4784510">goes</span> <span m="4784860">up</span> <span m="4785270">to</span> <span m="4785680">the</span> <span m="4786330">red</span> <span m="4786770">network</span> <span m="4788620">and</span> <span m="4789070">input</span> <span m="4789460">six</span> <span m="4790110">goes</span> <span m="4790430">straight</span> <span m="4791020">ahead</span> <span m="4791830">to the</span> <span m="4792050">blue</span> <span m="4792400">network.</span> <span m="4792680">It's</span> <span m="4792960">colored</span> <span m="4793200">blue</span> <span m="4793580">and</span> <span m="4793960">input</span> <span m="4794120">seven</span> <span m="4794370">is</span> <span m="4794480">also</span> <span m="4794740">colored</span> <span m="4795020">blue.</span> </p>
<p><span m="4797280">Let's</span> <span m="4797520">look</span> <span m="4797750">at</span> <span m="4797930">the</span> <span m="4798190">outputs.</span> <span m="4799330">So</span> <span m="4799740">for</span> <span m="4799970">example,</span> <span m="4802870">well,</span> <span m="4803100">let's</span> <span m="4803340">have a</span> <span m="4803570">look</span> <span m="4804010">at</span> <span m="4805030">output</span> <span m="4806140">zero.</span> <span m="4809700">so</span> <span m="4809930">output</span> <span m="4810280">zero--</span> <span m="4813210">which</span> <span m="4813740">packet</span> <span m="4814030">is</span> <span m="4814400">mapped</span> <span m="4814560">to</span> <span m="4814810">output</span> <span m="4814890">zero?</span> <span m="4815650">It's</span> <span m="4815830">packet</span> <span m="4816580">number</span> <span m="4817390">six.</span> <span m="4819570">So</span> <span m="4819860">six</span> <span m="4820490">was</span> <span m="4820760">mapped</span> <span m="4821580">into</span> <span m="4821890">the</span> <span m="4822020">blue</span> <span m="4822290">network</span> <span m="4823830">and</span> <span m="4824030">then</span> <span m="4824190">it</span> <span m="4824310">needs</span> <span m="4824550">to</span> <span m="4824650">be</span> <span m="4824810">mapped</span> <span m="4825320">to</span> <span m="4826520">output</span> <span m="4826870">zero.</span> <span m="4827300">So there's</span> <span m="4827440">only</span> <span m="4828040">one</span> <span m="4828360">edge</span> <span m="4828890">that</span> <span m="4829130">goes</span> <span m="4829380">from</span> <span m="4829570">the</span> <span m="4829670">blue</span> <span m="4829950">network</span> <span m="4831170">to</span> <span m="4831390">output</span> <span m="4831730">zero,</span> <span m="4833080">which</span> <span m="4833350">is</span> <span m="4834860">this</span> <span m="4835090">particular</span> <span m="4835570">one.</span> <span m="4836600">And</span> <span m="4836810">then</span> <span m="4836930">somehow,</span> <span m="4837750">this</span> <span m="4838050">one</span> <span m="4838200">needs</span> <span m="4838470">to</span> <span m="4838590">be</span> <span m="4838800">mapped</span> <span m="4840090">to</span> <span m="4840240">this</span> <span m="4840510">one</span> <span m="4840700">over</span> <span m="4840950">here.</span> </p>
<p><span m="4842490">Now,</span> <span m="4842610">we</span> <span m="4842720">can</span> <span m="4842870">continue</span> <span m="4843300">like</span> <span m="4843520">this.</span> <span m="4844800">Output</span> <span m="4845520">one</span> <span m="4846040">should</span> <span m="4846320">receives</span> <span m="4846700">a</span> <span m="4846810">packet</span> <span m="4847200">from--</span> <span m="4847610">let's</span> <span m="4847730">look</span> <span m="4847860">at</span> <span m="4847900">the</span> <span m="4848040">permutation--</span> <span m="4848700">from</span> <span m="4849320">five.</span> <span m="4850710">No,</span> <span m="4850940">sorry,</span> <span m="4851380">output</span> <span m="4851990">one--</span> <span m="4853520">pi of</span> <span m="4853820">0 is</span> <span m="4854270">equal</span> <span m="4854480">to</span> <span m="4854600">1</span> <span m="4855210">so</span> <span m="4855630">packet</span> <span m="4856040">zero</span> <span m="4856540">needs</span> <span m="4856770">to</span> <span m="4856880">go</span> <span m="4857140">to</span> <span m="4857790">this</span> <span m="4857960">particular</span> <span m="4858350">output.</span> <span m="4858770">Now,</span> <span m="4858950">packet</span> <span m="4859600">zero</span> <span m="4860360">is</span> <span m="4860710">in the</span> <span m="4860975">red</span> <span m="4861240">network</span> <span m="4862260">so</span> <span m="4862390">there's</span> <span m="4862510">only</span> <span m="4862850">one</span> <span m="4863120">edge</span> <span m="4864020">that</span> <span m="4864250">goes</span> <span m="4864940">from</span> <span m="4865150">the</span> <span m="4865250">red</span> <span m="4865530">network</span> <span m="4866020">to</span> <span m="4866080">this</span> <span m="4866290">output.</span> <span m="4867590">So</span> <span m="4868010">we</span> <span m="4868120">need</span> <span m="4868290">to</span> <span m="4868370">have a</span> <span m="4868540">connection</span> <span m="4869010">over</span> <span m="4869260">here.</span> </p>
<p><span m="4870710">Now,</span> <span m="4870900">we</span> <span m="4871010">can</span> <span m="4871160">continue</span> <span m="4871690">this</span> <span m="4872790">and</span> <span m="4873910">note</span> <span m="4874610">and</span> <span m="4874760">demonstrate--</span> <span m="4875450">and</span> <span m="4875580">you</span> <span m="4875740">can</span> <span m="4875960">test</span> <span m="4876240">it</span> <span m="4876340">for</span> <span m="4876530">yourself,</span> <span m="4877000">too--</span> <span m="4877970">that</span> <span m="4878260">output</span> <span m="4878690">four</span> <span m="4879250">needs</span> <span m="4879390">to</span> <span m="4879530">receive</span> <span m="4879880">a</span> <span m="4879970">packet</span> <span m="4880310">from</span> <span m="4880540">the</span> <span m="4880640">red</span> <span m="4880900">network.</span> <span m="4881890">Actually,</span> <span m="4882200">it</span> <span m="4882280">should</span> <span m="4882480">be</span> <span m="4883060">this</span> <span m="4883270">particular</span> <span m="4883790">one,</span> <span m="4884700">which</span> <span m="4884920">happens</span> <span m="4885180">to</span> <span m="4885260">be</span> <span m="4885420">packet</span> <span m="4885710">number</span> <span m="4885980">two.</span> <span m="4887580">And</span> <span m="4887680">then,</span> <span m="4887790">we</span> <span m="4888050">have</span> <span m="4888340">this</span> <span m="4888710">one,</span> <span m="4891060">right?</span> </p>
<p><span m="4892790">So</span> <span m="4893030">let</span> <span m="4893130">me</span> <span m="4893220">just</span> <span m="4893550">finish</span> <span m="4893990">it.</span> <span m="4894810">We</span> <span m="4894920">have</span> <span m="4895140">this</span> <span m="4896600">and</span> <span m="4896900">we</span> <span m="4897040">have</span> <span m="4899530">these</span> <span m="4899890">two</span> <span m="4901930">and</span> <span m="4902290">we</span> <span m="4902480">have</span> <span m="4904540">this</span> <span m="4904810">one.</span> <span m="4907350">We</span> <span m="4907580">have</span> <span m="4908010">this</span> <span m="4908320">one</span> <span m="4909620">and</span> <span m="4910060">we</span> <span m="4910310">have</span> <span m="4911460">this</span> <span m="4911750">one.</span> <span m="4913970">This</span> <span m="4914150">one</span> <span m="4914270">goes</span> <span m="4914580">straight</span> <span m="4915000">ahead.</span> <span m="4917480">This</span> <span m="4917760">one</span> <span m="4918170">goes</span> <span m="4918870">all</span> <span m="4919110">the</span> <span m="4919230">way</span> <span m="4919580">up</span> <span m="4920520">and</span> <span m="4920700">this</span> <span m="4920850">one</span> <span m="4920960">goes</span> <span m="4921220">all</span> <span m="4921390">the</span> <span m="4921460">way up.</span> </p>
<p><span m="4921920">So</span> <span m="4921980">what</span> <span m="4922100">do</span> <span m="4922210">we</span> <span m="4922340">see?</span> <span m="4923120">We</span> <span m="4923220">see</span> <span m="4923310">that</span> <span m="4923560">packets</span> <span m="4924750">over</span> <span m="4925080">here,</span> <span m="4926940">that</span> <span m="4927320">these</span> <span m="4928090">switches</span> <span m="4928470">only</span> <span m="4928750">see</span> <span m="4929020">a packet</span> <span m="4929350">once</span> <span m="4929860">and</span> <span m="4930020">these</span> <span m="4930250">ones,</span> <span m="4930460">as</span> <span m="4930790">well,</span> <span m="4931180">these</span> <span m="4931400">ones</span> <span m="4931640">also</span> <span m="4931905">and</span> <span m="4932170">these</span> <span m="4932510">ones</span> <span m="4932960">also.</span> <span m="4934100">So</span> <span m="4936090">we</span> <span m="4936240">have</span> <span m="4936650">directed</span> <span m="4937760">the</span> <span m="4937900">packets,</span> <span m="4938460">routed</span> <span m="4938870">the</span> <span m="4938970">packets</span> <span m="4939890">to</span> <span m="4940310">the</span> <span m="4940460">red</span> <span m="4940780">and</span> <span m="4941070">the</span> <span m="4941380">blue</span> <span m="4941820">subnetworks</span> <span m="4942860">in</span> <span m="4942950">such</span> <span m="4943280">a</span> <span m="4943380">way</span> <span m="4944050">that</span> <span m="4944360">the</span> <span m="4944470">congestion</span> <span m="4945220">at</span> <span m="4947160">the</span> <span m="4947280">last</span> <span m="4947660">level</span> <span m="4948160">and</span> <span m="4948440">at</span> <span m="4948680">the</span> <span m="4948770">first</span> <span m="4949160">level</span> <span m="4949920">is</span> <span m="4950110">still</span> <span m="4950470">equal</span> <span m="4950720">to</span> <span m="4950860">one.</span> </p>
<p><span m="4951680">Now,</span> <span m="4951940">we</span> <span m="4952050">use</span> <span m="4952300">our</span> <span m="4952460">induction</span> <span m="4952880">hypothesis</span> <span m="4954150">and</span> <span m="4954310">we</span> <span m="4954430">conclude</span> <span m="4955250">that</span> <span m="4955410">we</span> <span m="4955520">can</span> <span m="4955810">map</span> <span m="4958480">the</span> <span m="4958780">route</span> <span m="4959050">that's</span> <span m="4959450">going</span> <span m="4959560">to</span> <span m="4959600">have</span> <span m="4959750">a</span> <span m="4959830">routing</span> <span m="4960210">from</span> <span m="4960390">packets</span> <span m="4960830">from</span> <span m="4961000">here</span> <span m="4961240">to</span> <span m="4961450">here</span> <span m="4962220">such</span> <span m="4962490">that</span> <span m="4962590">the</span> <span m="4962690">congestion</span> <span m="4963380">within</span> <span m="4963730">the</span> <span m="4963820">subnetworks</span> <span m="4964690">is</span> <span m="4964930">only</span> <span m="4965250">one,</span> <span m="4966480">so</span> <span m="4966760">within</span> <span m="4967050">the</span> <span m="4967150">blue</span> <span m="4967490">as</span> <span m="4967580">well</span> <span m="4967960">as in</span> <span m="4968640">the</span> <span m="4968840">red.</span> <span m="4970930">So</span> <span m="4971130">this</span> <span m="4971370">is</span> <span m="4971680">the</span> <span m="4971850">insight</span> <span m="4972350">into</span> <span m="4972650">how</span> <span m="4972860">this</span> <span m="4973040">works</span> <span m="4974200">and</span> <span m="4975250">I</span> <span m="4975340">notice</span> <span m="4975530">I</span> <span m="4975790">am</span> <span m="4976060">running</span> <span m="4976370">out</span> <span m="4976540">of</span> <span m="4976660">time.</span> <span m="4976950">So</span> <span m="4977130">the</span> <span m="4977230">formal</span> <span m="4977660">proof</span> <span m="4978300">we</span> <span m="4978380">will</span> <span m="4978480">have</span> <span m="4978670">to</span> <span m="4978790">postpone</span> <span m="4979400">until</span> <span m="4979920">recitation,</span> <span m="4981460">but</span> <span m="4983130">that's</span> <span m="4983610">actually</span> <span m="4984120">really</span> <span m="4984640">a</span> <span m="4984760">very</span> <span m="4985030">simple</span> <span m="4985540">thing</span> <span m="4985890">to</span> <span m="4986010">do</span> <span m="4986170">that</span> <span m="4986280">right</span> <span m="4986620">now.</span> </p>
<p><span m="4987540">So</span> <span m="4987690">just</span> <span m="4988030">keep</span> <span m="4988250">this</span> <span m="4988570">key</span> <span m="4988830">insight</span> <span m="4989670">and</span> <span m="4989910">then</span> <span m="4991300">you</span> <span m="4991430">can</span> <span m="4991640">easily</span> <span m="4992870">prove</span> <span m="4993520">the</span> <span m="4993770">theorem.</span> <span m="4995910">But</span> <span m="4996020">this</span> <span m="4996190">is</span> <span m="4996350">the</span> <span m="4996460">real</span> <span m="4997490">insight.</span> <span m="4998820">Thank</span> <span m="4999120">you.</span> </p>
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